Convex functions

SCIENTIFIC AND INDUSTRIAL NEWS

PRESENTATIONS ON THE THEORY OF FUNCTIONS

Published under the direction of
Paul MONTEL,
Member of the Institute,
Professor at the Faculty of Sciences of the University of Paris

THE

CONVEX FUNCTIONS

BY

TIBERIU POPOVICIU

Professor at the University of Iași

PARIS
HERMANN & $\mathrm{C}^{\mathrm{i}}$ , PUBLISHERS
6, Rue de la Sorbonne, 6
solei
1944

INTRODUCTION

Since Jensen wrote these lines in his now classic memoir [27] $({}^{1})$ The importance of the concept of convexity has increased considerably in many branches of modern mathematics, and, in particular, in geometry and function theory.

In this short book, I present the main properties and some generalizations of convex functions of one or more variables. I have divided this work into four chapters. In the first, I study the theory of order functions. $n$ of which the usual convex functions are a special case $(n=1)$ Order functions $n$ are defined by certain inequalities and verify others which have, especially for $n=1$ , numerous applications. In the second chapter, I briefly review these inequalities, the applications of which are set forth in excellent works, such as, for example, Inequalities by Messrs. Hardy, Littlewood, and Pólya [21 h]. I also point out some other properties of order functions $n$ In the third chapter, I examine some generalizations of order functions. $n$ The fourth chapter is devoted to convex functions of two or more variables. I limit myself almost exclusively to the case of two variables because, on the one hand, the simplest properties extend immediately to the case of more than two variables and, on the other hand, the more complicated properties have not yet been studied sufficiently.

⁰⁰footnotetext: ⁽¹⁾ Les nos en caractères gras entre crochets renvoient à la bibliographie placée à la fin.

The bibliography is not intended to be exhaustive. I only list the works actually used in writing this book.

As usual, I do not provide the proofs; the reader will find them in the original memoirs to which I refer. I briefly indicate the proofs of some properties that are not found in these memoirs. For all other definitions and properties without indications of the proof or references to the bibliography, I ask the reader to refer to my Thesis [47 a].

I hope that reading this short work will be useful to those who seek to fill the still very numerous gaps in this theory.

Allow me to express my deepest gratitude to Mr. Paul Montel for the honor he bestowed upon me by asking me to write this book on a subject which he himself has enriched with important contributions.

PRELIMINARY CONCEPTS AND NOTATIONS

We consider functions $f=f(x)$ , real, of the real variable $x$ , finite and uniform on any linear set $E$ .

We will refer to $a=\min E$ , $a\leqq b=\max E$ the extremities (left and right) of $E$ When we say that a set is closed, we assume that it contains its endpoints, and therefore that it is bounded. To simplify, we write $x,y,\ldots\in E$ instead of $x\in E,y\in E,\ldots$ , and we write $x\in E$ if the point $x$ of $E$ is within the interval $(a,b)$ A subset $E_{1}$ of $E$ is completely internal to $E$ if its endpoints are within the interval $(a,b)$ Such a subset is therefore always bounded. In this case, we write $E_{1}\subset E$ A subset $E_{1}$ of $E$ is a section of $E$ if, either it is formed by a single point, or else, with $x_{1},x_{2}\in E_{1}$ , all the points of $E$ belonging to the interval $(x_{1},x_{2})$ belong to $E_{1}$ If a section contains its endpoints $c\leqq d$ we will also refer to it as $(cEd)$ The intersection of two sections is either empty or a section of $E$ $({}^{1})$ The union of two sections having at least one point in common is still a section. Two sections whose union does not constitute a section are said to be separated by $E$ ; then there exists at least one point of $E$ which is to the left of all points in one section and to the right of all points in the other section. Several sections of $E$ are separate sections of $E$ if they are separated in pairs by $E$ We will refer to it, as usual, as $E^{\prime},E^{\prime\prime},\ldots$ the successive derivatives of $E$ The near-closure $\dot{E}$ of $E$ is the set of all points of $E$ and $E^{\prime}$ except for the extremities of $E$ that do not belong to $E$ . If $\dot{E}=E$ , we will say that the whole $E$ is almost closed.

We will say that a function $f$ is continuous on a set $E$ if it is continuous at every point of $E^{\prime}$ We will say that $f$ has a derivative (of a certain order and with a certain definition) if this derivative exists at every point $x$ where it is possible to define it, and we will say that $f$ has a continuous derivative on $E$ if this derivative is continuous on its domain.

We will refer to $[x_{1},x_{2},\ldots,x_{n+1};f]$ the difference divided by order $n$ of $f$ on the points $x_{1},x_{2},\ldots,x_{n+1}$ These divided differences are defined by the recurrence relation.

,\ldots,x_{n+1};f\begin{gathered}=\frac{[x_{2},x_{3},\ldots,x_{n+1};f]-[x_{1},x_{2},\ldots,x_{n};f]}{x_{n+1}-x_{1}},\\ =f(x).{}\end{gathered}

and we can see that they are symmetrical with respect to the points $x_{1},x_{2},\ldots,x_{n+1}$ Let us designate by $V(x_{1},x_{2},\ldots,x_{n+1})$ the Vandermonde determinant of numbers $x_{1},x_{2},\ldots,x_{n+1}$ and by $\mathrm{U}(x_{1},x_{2},\ldots,x_{n+1};f)$ what happens to this determinant if we replace the elements $x_{i}^{n}$ from the last column by $f(x_{i})$ respectively. We have

[x_{1},x_{2},\ldots,x_{n+1};f]=\frac{\mathrm{U}(x_{1},x_{2},\ldots,x_{n+1};f)}{\mathrm{V}(x_{1},x_{2},\ldots,x_{n+1})}.

We will refer to $\mathrm{P}(x_{1},x_{2},\ldots,x_{n+1};f\mid x)$ the Lagrange polynomial, therefore the polynomial of minimum effective degree, taking the values $f(x_{i})$ at the points $x_{i}$ It is a polynomial of degree $n$ , by agreeing to call by this name a polynomial of effective degree $\leqq n$ If we ask $\varphi(x)=(x-x_{1})(x-x_{2})\ldots(x-x_{n+1})$ , We have

		$\displaystyle=\sum_{i=1}^{n+1}\frac{f(x_{i})}{\varphi^{\prime}(x_{i})},$
	$\displaystyle P(x_{1},x_{2},\ldots,x_{n+1};f\mid x)$	$\displaystyle=\sum_{i=1}^{n+1}\frac{\varphi(x)f(x_{i})}{(x-x_{i})\varphi^{\prime}(x_{i})}.$

And

f(x)-\mathrm{P}(x_{1},x_{2},\ldots,x_{n+1};f\mid x)=\varphi(x)[x_{1},x_{2},\ldots,x_{n+1},x;f].

Unless otherwise stated, or unless the writing expressly indicates otherwise, we will assume in these notations that $x_{1}<x_{2}<\ldots<x_{n+1}$ and then we have $\mathrm{V}(x_{1},x_{2},\ldots,x_{n+1})>0$ , SO

\operatorname{sg}[x_{1},x_{2},\ldots,x_{n+1};f]=\operatorname{sg}\mathrm{U}(x_{1},x_{2},\ldots,x_{n+1};f),

( $\operatorname{sg}\alpha=1,0,-1$ depending on $\alpha=,>,<0$ ).
Let us also note the important property

[x_{1},x_{2},\ldots,x_{n+1};x^{m}]=\begin{cases}0,&m=0,1,\ldots,n-1,\\ 1,&m=n,\end{cases}

identically in the $x_{i}$ and which characterizes divided differences. In Chapter IV, we consider functions $f(x,y)$ real, real variables $x,y$ , finite and uniform within a certain planar domain $D$ We will always assume that $D$ is a bounded convex domain, closed or not. A subdomain of $D$ is completely internal to $D$ if its border has no point in common with the border of $D$ We will, moreover, always assume that the subdomain under consideration is also convex. More specifically, we will assume that $D$ is a closed rectangle $R$ ,

(a\leqq x\leqq b,\quad c\leqq y\leqq d).

According to A. Marchaud [36], a network of order $(m,n)$ is a system of $m$ parallel to the axis $\mathrm{O}y$ and $n$ parallel to the axis $\mathrm{O}x$ The points of intersection of the lines in a network are the nodes of that network. Following a denomination by A. Marchaud [36], a pseudo-polynomial of order $(m,n)$ is a function of the form

\sum_{i=0}^{m}x^{i}A_{i}(y)+\sum_{j=0}^{n}y^{j}B_{j}(x),

Or $A_{i}(y)$ are functions of a variable $y$ In $(c,d)$ And $B_{j}(x)$ functions of a variable $x$ In $(a,b)$ A pseudo-polynomial is therefore defined in the rectangle $R$ A pseudo-polynomial of order $(m,n)$ is completely determined by its values on an order network $(m+1,n+1)$ Consider $(m+1)(n+1)$ points $(x_{i},y_{j})$ , $i=1,2,\ldots,m+1$ , $j=1,2,\ldots,n+1$ , of $R$ These are the nodes of the network. $x=x_{i},y=y_{j}$ , of order $(m+1,n+1)$ By taking the difference divided by order $m$ of $f(x,y)$ on the points $x_{1},x_{2},\ldots,x_{m+1}$ , $y$ being considered fixed, we have the function of $y$ ,

F(y)=[x_{1},x_{2},\ldots,x_{m+1};f(x,y)],

and taking the difference divided by order $n$ of $f(x,y)$ on the points $y_{1},y_{2},\ldots,y_{n+1}$ , $x$ being considered fixed, we have the function of $x$ ,

G(x)=[y_{1},y_{2},\ldots,y_{n+1};f(x,y)].

We immediately check that

[y_{1},y_{2},\ldots,y_{n+1};F]=[x_{1},x_{2},\ldots,x_{m+1};G].

The common value of these numbers can be designated by

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\right]

and is called the divided difference of order $(m,n)$ of $f(x,y)$ on the points $(x_{i},y_{j})$ We will ask

	$\displaystyle\varphi(x)$	$\displaystyle=(x-x_{1})\cdots(x-x_{m+1}),$
	$\displaystyle\psi(y)$	$\displaystyle=(y-y_{1})\cdots(y-y_{n+1}),$

and we can write

\left[\begin{array}[]{l}x_{1},\ldots,x_{m+1}\\ y_{1},\ldots,y_{n+1};f\end{array}\right]=\sum_{i=1}^{m+1}\sum_{j=1}^{n+1}\frac{f(x_{i},y_{j})}{\varphi^{\prime}(x_{i})\psi^{\prime}(y_{j})}.

The difference divided by order $(m+1,n+1)$ of a pseudo-polynomial of order $(m,n)$ is null and void. Unless otherwise stated, we will assume that $x_{1}<\cdots<x_{m+1},y_{1}<\cdots<y_{n+1}$ We
will refer to it as $[\alpha]$ the largest whole $\leqq\alpha$ We
will ask $(m_{1},n_{1})<\text{ resp. }\leqq(m,n)$ if $m_{1}\leqq m,n_{1}\leqq n$ And $m_{1}+n_{1}<\text{ resp. }\leqq m+n$ Finally, we use the abbreviations max and min for the upper and lower bounds, and dd for the divided difference.

CHAPTER I

ORDER FUNCTIONS $n$

1. Definition. The function $f$ is said to be convex, non-concave, polynomial, non-convex, concave of order $n$ on $E$ if inequality

[x_{1},x_{2},\ldots,x_{n+2};f]>,\geq,=,\leqq,<0

(1)

is satisfied, whatever the $n+2$ points $x_{1},x_{2},\ldots,x_{n+2}\in E$ All these functions are functions of order $n$ $({}^{1})$ . For $n=0$ We have monotonic functions: increasing, non-decreasing, constant, non-increasing, and decreasing. $n=1$ We have convex, non-concave functions. If $f$ is convex, non-concave, etc., of order $n$ , the function $-f$ is concave, non-convex, etc., of order $n$ , and vice versa. Let's consider the function $f$ defined on $m$ points

x_{1}<x_{2}<\cdots<x_{m},\quad m\geq n+2.

(2)

and let's use the notation

\Delta_{k}^{i}=[x_{i},x_{i+1},\ldots,x_{i+k};f],\quad i=1,\ldots,m-k.

(3)

\Delta_{n+1}^{i}>,\geq,=,\leqq,<0.

(4)

\mathrm{P}(x_{1},x_{2},\ldots,x_{n+1};f\mid x)<,\leqq,=,\geqq,>f(x),

(5)

is verified, regardless of $x_{n+1}<x,\quad x_{1},x_{2},\ldots,x_{n+1},x\in\mathbf{E}$ This
definition is equivalent to the definition of $n^{0}1$ We
can replace inequality (5) with

\begin{gathered}\mathrm{P}(x_{1},x_{2},\ldots,x_{n+1};f\mid x)<,\leqq,\cdots,>(-1)^{n-i+1}f(x)\\ x_{i}<x<x_{i+1}\ (x<x_{1}\text{ pour }i=0),\quad x_{1},x_{2},\ldots,x_{n+1},x\in E.\end{gathered}

For $n=1$ see L. Galvani [19]. In this case, limiting ourselves to non-concave functions, the property means that the point $(x,f(x))$ is neither above nor below the line joining the points $(x_{1},f(x_{1})),(x_{2},f(x_{2}))$ depending on $x$ is inside or outside the interval $(x_{1},x_{2})$ It follows that every point on the curve $y=f(x)$ is not above any inscribed polygonal line, provided that the abscissa of this point is included between the abscissas of the extreme points of the polygonal line.

If $f$ is of order $n$ and if $[x_{1},x_{2},\ldots,x_{n+2};f]=0$ , it is a polynomial of order $n$ on $(x_{1}Ex_{n+2})$ , therefore reduces in this section to the values of a polynomial of degree $n$ .

A convex or concave function of order $n$ cannot coincide with a polynomial of degree $n$ in addition to $n+1$ points. The converse is true if the function is continuous.

Suppose that $E$ let the interval $(a,b)$ and either $F$ the function attached to $f$ , taking into account $x$ all values between the minimum and maximum of $f$ at this point (this function is generally multifaceted). So that the function $f$ either convex or concave of order $n$ In $(a,b)$ It is necessary, and sufficient, that the attached function $F$ does not coincide with a polynomial of degree $n$ in addition to $n+1$ points $[47\mathrm{g}]$ In general, order functions $n$ can be characterized by the fact that, if a polynomial of degree $n$ coincides with the attached function $F$ in addition to $n+1$ points, it coincides with $F$ in a closed interval.
3.—Any function of order $n$ is bounded along any section $E_{1}\subset E$ . If $a,b\in\mathbf{E}$ the function is bounded on $E$ . When $E$ is bounded, every non-concave function of odd order is bounded below on $E$ .

For a function of order $n\geq 1$ And $c\in E^{\prime},c\in E$ , the limit of $f(x)$ For $E\ni x\to c$ exists. Moreover, any order function $n\geq 1$ is uniformly continuous over all $E_{1}\subset E$ For what we have said to have precise meaning, we must define continuity in $c\in E^{\prime}$ by the existence of the limit of $f(x)$ when $E\ni x\to c$ , which is not an essential extension in the case of order functions $\geq 1$ .

If $a\in E^{\prime}$ Or $a=-\infty$ , $\lim f(x)=f(a+0)$ For $E\ni x\to a$ , exists or is $+\infty$ Or $-\infty$ . If $a\in E$ we have $f(a)\leqq$ Or $\geqq f(a+0)$ depending on $f$ is non-concave of even or odd order. A similar property holds true for the right end $b$ We have $f(b)\geq$ Or $\leqq f(b-0)$ depending on $f$ is non-concave or non-convex of order $n$ For a non-concave function of order $1$ such as $b=+\infty$ And $\lim f(x)=+\infty$ For $E\ni x\to b$ we can find a number $\alpha>0$ such as $f(x)>\alpha x$ For $x$ big enough. If $f$ is non-concave of order $n$ and if it is not non-convex of order $n-1$ on $E$ , one can find, under the same conditions, a $\alpha>0$ such as $f(x)>\alpha x^{n}$ Indeed, one can then find $n+1$ points $x_{1}<x_{2}<\ldots<x_{n+1}$ such as $[x_{1},x_{2},\ldots,x_{n+1};f]>0$ and the inequality of definition $\mathrm{U}(x_{1},x_{2},\ldots,x_{n+1},x;f)\geqq 0,\ x_{n+1}<x$ gives us

f(x)\geq[x_{1},x_{2},\ldots,x_{n+1};f]x^{n}+\cdots,

the unwritten terms forming a polynomial of degree $n-1$ in $x$ 4.
—Suppose $E$ closed. Any non-concave function of order $n$ The odd function is semi-continuous in the upper bound, and therefore reaches its maximum. However, such a function may not reach its minimum. This is the case, for example, with the function $f(0)=1,f(x)=x,0<x\leqq 1$ , which is non-concave of any odd order in $(0,1)$ A function of even order may never reach its maximum or minimum. Such is the function $f(0)=f(2)=0,f(x)=1-x,0<x<2$ , which is of any even order in $(0,2)$ .

Let $E_{M}$ all of $x$ where max $f$ on $E$ And $E_{m}$ all of $x$ where min $f$ on $E$ are reached. If $f$ is non-concave of order $n$ , $E_{M}$ is formed by at most $\left[\frac{n+3}{2}\right]$ separate sections of $E$ and, if it is not formed by a single section, it contains at most $n+1$ points.

5.—Definition

The function $f$ enjoying certain convexity properties on $E$ , will be said to be extendable on $E_{1}$ if we can find a function $f_{1}$ defined on the meeting of $E$ And $E_{1}$ , enjoying the same convexity properties and such that $f_{1}=f$ on $E$ .

presents at most $k$ sign variations.
For a function of order $n$ and for a given k there exists an infinite number of decompositions (7) (E being infinite), such that on each $\mathrm{E}_{i}$ the function is of order $n-k$ The number $m$ subsets $\mathrm{E}_{i}$ then has a minimum $h$ . If $m=h$ the function is not of order $n-k$ on none of the sets $\mathrm{E}_{i}+\mathrm{E}_{i+1},i=1,2,\ldots,h-1$ The function is then alternately non-concave and non-convex of order $n-k$ on the sets $\mathrm{E}_{i}$ Moreover, if $h=k+1$ and if $f$ is nonconcave of order $n$ on E, it is non-concave of order $n-k$ on $\mathbf{E}_{n}$ .

Conversely, for the function to be of order $n$ on $\mathbf{E}$ , it suffices that, whatever the polynomial $P$ degree $n$ and the finite subset (2) of $E$ , the sequence (8), corresponding to (2) and to the function $f-P$ , presents at most $n$ sign variations. In this statement, we can also consider only the sequences (2) having $n+3$ points $[47\,k]$ .

Functions that admit a decomposition (7) of the preceding nature constitute an important generalization of order functions $n$ We will study them in another work. Let us only note that [47 n].

The necessary and sufficient condition for decomposing E into at most two consecutive subsets such that on each the function is monotonic, with the monotonicity being in opposite directions on the two subsets, is that $f$ Or - $f$ checks the inequality

f\left(x_{2}\right)\leq\max\left[f\left(x_{1}\right),f\left(x_{3}\right)\right],\quad x_{1}<x_{2}<x_{3},\quad x_{1},x_{2},x_{3}\in\mathrm{E}.

This class includes not only first-order functions, but also all non-negative functions whose pth power, $p>1$ , is of order 1.
8. - The neighborhood $V_{x}^{k}$ from one point $x$ is a section of $E$ having at least $k$ points on the left and at least $k$ points to the right of $x$ If, exceptionally, there is only $r<k$ ( $r\geq 0$ ; points to the left (right)
of $x,\mathrm{~V}_{x}^{k}$ must contain all these points and at least $2k-r$ points to the right (left) of $x$ The neighborhood $V_{a}^{k}$ must be consistent with $x_{1}\in\mathrm{~V}_{a}^{k}$ all points of the closed interval ( $a,x_{1}$ ) belonging to the EU, the same applies to a $\mathrm{V}_{b}^{k}$ .

Definition. - The function $f$ is said to be locally convex, nonconcaceous, ..., etc., of order n on E if at all $x\in\dot{E}$ corresponds to a poisinage $\mathrm{V}_{x}^{k}$ where the function is convex, non-concave,… etc., of order n [47 1].

Any locally convex, non-concave, etc. function of order $n$ on N, with $k=\left[\frac{n+2}{2}\right]$ , is convex, non-concave, … ote., of order n on E [47 1].

The condition cannot be replaced in the definition $x\in\dot{E}$ by the less restrictive condition $x\in$ E. For example, with this new definition, the function

f(x)=0,\quad 0\leqq x<1,\quad f(x)=x-1,\quad 1<x\leq 2

is locally polynomial of order $n\geq 1$ (for any k) and yet this function is not of order $n$ .

In the case of an interval, we can replace $V_{r}^{h}$ by a neighborhood with ordinary sounds. For $n=1$ we find a property of J. Blaquier [8].

One can also impose on a neighborhood $V_{x}^{k}$ other conditions leading to convexity. We can say that / locally has a line of support on E if, for all $x_{0}=$ E, there exists a neighborhood $V_{x_{0}}^{1}$ and a non-vertical line passing through the point ( $\left.x_{0},f\left(x_{0}\right)\right)$ leaving the curve $y=f(x)$ no, below this line for $a\in\mathbb{V}_{x_{0}}^{t}$ Therefore, any function defined and continuous on the almost formed set E, which locally has a support line, is non-concave of order 1 on E. We can even prove the following property:

If $f$ is defined and semi-continuous in a superior manner on an almost formed set E ot if, whatever $x_{0}\equiv$ E, we can find two points $x^{\prime}<x_{0}<x^{\prime\prime}$ such as in any neighborhood $V_{x_{0}}^{1}\subset\left(x^{\prime},x^{\prime\prime}\right)$ there are two points $x_{1},x_{2},x_{1}<x_{0}<x_{2}$ verifying the inequality $\left[x_{0},x_{1},x_{2};f\right]\geqq 0$ , the function is non-concave of order 1 on $E$ [471].

For the demonstration, it suffices to note that, if $f$ is not non-concave of order 1, we can find three points $x_{1},x_{2},x_{3}$ , $x_{1}<x_{2}<x_{3}$ of E so that $\left[x_{1},x_{2},x_{3};f\right]<0$ The extremities of the (closed) set on which the maximum $(>0)$ of the function $f(x)-\frac{x-x_{3}}{x_{1}-x_{3}}f\left(x_{1}\right)-\frac{x-x_{1}}{x_{3}-x_{1}}f\left(x_{3}\right)$ on the section $\left(x_{1}\mathrm{E}x_{2}\right)$ is waiting, are points $x_{0}\in$ E for which points cannot be found $x^{\prime},x^{\prime\prime}$ satisfactory to the property requirements.

The property of being of order n is not a local property. But, if at all $x\in\dot{E}$ corresponds to a neighborhood $V_{x}^{k}$ , with $k=\left[\frac{n+3}{2}\right]$ , Or $f$ is convex or concave of order $n$ , this function is convex or concave of order $n$ on E [471].
9. - In the study of order functions $n$ It is quite appropriate to introduce two other classes of functions already considered by E. Hopp [23] in the case of an interval.

Definition. - The nth bound of the function f on E is defined by

\Delta_{n}=\Delta_{n}[f;\mathbf{E}]=\max_{(\mathbb{E})}\left|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\right|

If $\Delta_{n}$ is a finite number the function is said to have nth difference, divisible and bounded on E.

Definition. - The nene total parity of the function $f$ on $\mathbf{E}$ is defined by

V_{u}=V_{n}[f;E]=\max\sum_{i=1}^{m-1}\left|\Delta_{n}^{i+1}-\Delta_{n}^{i}\right|

the maximum being taken for all finite subsets (2) of E.
If $\mathrm{V}_{\text{a }}$ is a finite number the function is said to have nth parity bounded on E.

We will assume that E is bounded.
If $n=1$ , we have the functions satisfying an ordinary Lipschitz condition and a generalization of the functions with bounded variation already given by Ch. de la Vallée Poussin [62], F. Rhesz [50 a] eb A. Winternitz [65].

On the finite set (2), $\Delta_{n}$ coincides with the maximum of the
numbers $\left|\Delta_{n}^{1}\right|,\left|\Delta_{n}^{2}\right|,\ldots,\left|\Delta_{n}^{m-n}\right|$ From this it follows that, if $\mathbf{E}$ is arbitrary, one can find a $x_{0}$ the closure of $\mathbf{E}$ such as in any neighborhood $\mathrm{V}_{x_{0}}^{k}$ , with $k=\left[\frac{n+1}{2}\right]$ , we have $\Delta_{n}\left[f;\mathrm{V}_{x_{0}}^{lk}\right]=\Delta_{n}[f;\mathrm{E}]$ [47 1]. It follows that if $f$ is at nth dd bounded in the neighborhood (in the ordinary sense) of every point $x\in\mathrm{E}^{\prime}$ , it is at nth dd bounded on E.

Every bounded nth dd function is also a (n - 1)th dd bounded function, therefore, in particular, is bounded.

Every function with bounded nth variation is also with bounded nth variation dd and to $(n-1)^{\text{ème }}$ bounded variation. Similarly, any bounded function dd with nth d is at ( $n-1$ )th bounded variation.

Any order function $n$ is of bounded nth variation, therefore also of bounded nth variation dd on any section $\mathrm{E}_{1}\subset.\mathrm{E}$ Moreover, if $\Delta_{n}$ is finished, we can find a number $\alpha$ such as $f+\alpha x^{n}$ (For example $\alpha=\left|\Delta_{n}\right|$ ) either of order $n-1$ For this to be the case, it is sufficient that the order dd $n$ be bounded above or below.

Any function $f$ has $n^{\text{ème }}$ bounded variation is the difference between two non-concave functions of order $-1,0,1,\ldots,n$ and whose nth total variations do not exceed that of $f$ This result was obtained by E. Hopf [23] in the case of an interval and for $n=1$ by A. Winternitz [65]. For $n=0$ We obtain a classical theorem of C. Jordan on ordinary functions of bounded variation. In the general case, such a decomposition exists. $f=\varphi-\psi$ , where the functions $\varphi$ , $\psi$ are as small as possible. $n=0,1\mathrm{G}$ Ascoli [1 a, 1 b ] rediscovered these properties through very interesting considerations.
10. - We will now study the derivatives of order functions $n$ We will assume E is closed and $f$ bounded on E.

We will adopt a direct definition of the nth derivative, more restrictive than the usual definition, but which is necessary in the study of order functions. $n$ .

By definition, the $n^{eme}$ derivative $f^{(n)}=f^{(n)}(x)$ to the point $x\in\mathrm{E}^{\prime}$ is the limit, if it exists, of $n!\left[x_{1},x_{2},\ldots,x_{n+1};f\right]$ when the points $x_{i}$ tend in some way towards $x$ The derived term can thus be defined at any point of E' while the $n^{\text{ème }}$ derivative in the ordinary sense is defined only on the points of $\mathrm{E}^{(n)}$ .

So that $f^{(n)}$ exists at a point $x\in\mathrm{E}^{\prime}$ it is necessary and sufficient that at all $\varepsilon>0$ corresponds to a neighborhood $V$ (in the ordinary sense) such that one has

	$\displaystyle\left\|\left[x_{1},x_{2},\ldots,x_{n+1};f\right]-\left[x_{2},x_{3},\ldots,x_{n+2};f\right]\right\|<\varepsilon$		(9)
	$\displaystyle x_{1},x_{2},\ldots,x_{n+2}\in\mathrm{~V}.$

So that $f^{(n)}$ either continue on $\mathrm{E}^{\prime}$ It is necessary and sufficient that this condition be satisfied uniformly on E'. Moreover, if $f^{(n)}$ exists in every point of $\mathrm{E}^{\prime}$ it is continuous on $\mathrm{E}^{\prime}$ . If $f^{(n)}$ exists in one $x_{0}\in\mathrm{E}^{\prime}$ the function is named after horned in the vicinity of $x_{0}$ , done if $f^{(n)}$ exists at every point of E', the function $f$ is, at nth dd bomée on E. It is clear, on the other hand, that if $f$ is at ( $n+1$ )cane d. $d$ bounded on $E,f^{(n)}$ exists at every point of $E$ .

The relationships that exist between $f^{(n)}$ and the nth derivative in the ordinary sense have been studied in the case of an interval by Th. J. Stuelpes [56], E. Hopp [23], Ph. Franklin [17] and in the case of an arbitrary set E by ourselves. In particular, if $f^{(n)}$ exists at a point, $x_{0}\in\mathrm{E}^{(n)}$ the derivative of order $n$ ordinary sound also exists at this point and is equal to it.

We can also define a left-handed direct derivative of order n $f_{b^{\prime}}^{(\prime\prime)}$ and a direct derivative of order $n$ right $f_{d}^{(n)}$ to the point $x\in E_{5}^{\prime}$ By definition $f_{g}^{(n)}(x)\left[f_{d}^{(n)}(x)\right]$ is equal to the limit, if it exists, of $n!\left[x_{1},x_{2},\ldots,x_{n+1};/\right]$ when the points $x_{i}\leq x\left[x_{i}\geq x\right]$ of $E$ tend, in some way, towards $x$ For the existence of these derivatives, there are necessary and sufficient conditions analogous to that expressed by inequality (9). It is clear that if $x\in\mathrm{E}^{\prime}$ is limited only on one side, we only define the nth derivative of that side, which is then identical to $f^{(n)}$ . If $x\in\mathrm{E}^{\prime}$ is the limit on two sides of $f^{(n)}(x)$ Oxiste, $f_{g}^{(n)}(x),f_{l}^{(n)}(x)$ They also exist and are equal to it. $f_{g}^{(m)}(x),f_{d}^{(n)}(x)$ can exist and even be equal without $f^{(n)}(x)$ exists. For example, for the function $f(x)=|x|,x\in(-1,1)$ , $f_{g}^{\prime\prime}(0),f_{d}^{\prime\prime}(0)$ both exist and are both equal to zero, but $f^{\prime\prime}(0)$ does not exist. We also see that, if $f_{y}^{(\prime\prime)}\left[f_{d}^{(\prime\prime)}\right]$ exists at every point of E', e' is a left-continuous (right-continuous) function on E'.
11. – Let's move on to functions of order n. Every function of order $n>1$ has continuous derivatives of order $1,2,\ldots,n-1$ on mowing section-
tion $\mathrm{E}_{1}\in$ E. It can even be shown that, if $f$ is of order $n\geqq 1$ in the meantime ( $a,b$ ), it has continuous derivatives of order $\alpha<n(x>0)$ in the sense of Liouville-Riemann throughout $(c,d)\subset\cdot(a,b)$ This is, moreover, the case for any function to $n^{\text{òme }}$ bounded dd, as shown by P. Montel [39 a]. In this case, the derivatives of integer order exist and are continuous throughout the interval (a, b) and A. Marghaud [36] showed that this is also the case for derivatives of non-integer order.

Any order function $n>1$ has a left derivative of order $n$ and a right-hand derivative of order $n$ , continues to the left or right over the entire section $\mathrm{E}_{1}\subset$ E. Well understood $f_{g}^{(n)}\left(y_{d}^{(n)}\right)$ is not defined at a point $x\in\mathrm{E}^{\prime}$ which is limited only to the right (left), but then $f^{(n)}=f_{d}^{(n)}\left(f^{(n)}=f_{g}^{(n)}\right)$ exists at this point. To ensure consistency in what we say here, we can assume that $f_{g}^{(n)}\left(f_{l}^{(n)}\right)$ be extended by $f^{(n)}$ on these points. The existence of derivatives of order $n$ results from the fact that the dd $\left[x_{1},x_{2},\ldots,x_{n+1};f\right]$ is monotonic, of the same direction, with respect to each of the variables $x_{i}$ . Besides, $f_{g}^{(n)},f_{d}^{(n)}$ coincide with the left-hand derivative and the right-hand derivative of $f^{(n-1)}$ at one point $x\in\mathrm{E}^{\prime\prime}$ If at a point $x\in\mathrm{E}^{\prime}$ we have $f_{g}^{(n)}=f_{l}^{(n)}$ the derivative of order $n,f^{(n)}$ also exists and is equal to them. The derivative $f^{(n)}$ is continuous throughout $\mathrm{E}_{1}\subset\mathrm{E}$ on which it exists, as L. Galvani noted [19] for $n=1$ . If $x<x^{\prime}$ , $x,x^{\prime}\in\mathrm{E}^{\prime}$ we have

f_{l}^{(n)}(x)\leqq\left[x_{1},x_{2},\ldots,x_{n+1};f\right]\leqq f_{k^{\prime}}^{(n)}\left(x^{\prime}\right),\quad x_{1},x_{2},\ldots,x_{n+1}\in\left(x\left[x^{\prime}\right)\right.

And

f_{g}^{(n)}(x)\leqq f_{d}^{(n)}(x)\leqq f_{b}^{(n)}\left(x^{\prime}\right)\leqq f_{d}^{(n)}\left(x^{\prime}\right)

provided that $/$ either non-concave of order $n$ on E. The set of $x$ on which $f_{g}^{(n)}\neq f_{d}^{(n)}$ is moreover at most countable, as noted by F. Bernstein [5] and L. Galvant [19] for $n=1$ . For $n=1$ the existence of derivatives on both sides has already been established by O. Stolz [57] and JLWV Jensen [27].

If $f$ is non-concave of order $n,f^{(k)}$ is non-concave of order $n-k$ . Likewise, $f_{g}^{(n)},f_{d}^{(n)}$ are non-decreasing and $f^{(n+1)},f_{g}^{(n+1)}$ , $f^{(n+1}$ are non-negative wherever they exist.

It remains to examine what happens at the extremities of E. If
a EE la dd n I , $\left[x_{1},x_{2},\ldots,x_{n-1};f\right]$ tends towards a limit on yours $+\infty$ , Or $-\infty$ , if the $x_{i}-a$ tend towards a. If ae E' limit coincides with the limit, in the proper or improper sense, of $f_{11}^{(n)}$ and $f_{a}^{(n)}$ when $\mathbf{E}^{\prime}\Rightarrow x=a$ tends towards $a$ Furthermore, if the function is continuous at a and if the limit exists (in the proper or improper sense), $f_{a}^{(n)}(a)$ , done $f^{(n)}(a)$ exists (in the literal or literal sense) and is equal to the limit. It is clear, moreover, that this limit is ¡ + $\infty$ if the function is non-concave of order n and is - - os if the function is non-convex of order $n$ If the function is bounded in the neighborhood of a, then all derivatives $f^{(i)}(a),i=1,2,\ldots$ , $k$ exist and ceoi for $k\leqq n$ Similar remarks can be made regarding the extremity $b$ .

According to a remark (in the case $n=1$ (a hypothesis which, moreover, is not essential here) by W. Blasumke and C. Pick [9], if $f$ is of order 1 on E and if $a\in\mathrm{E}^{\prime\prime}$ we have

(x-a)f_{g}^{\prime}(x)\rightarrow 0,\quad(x-a)f_{d}^{\prime}(x)\rightarrow 0,

if $E^{\prime}\exists x=$ a tends towards $a$ Taking into account the results of the $n^{\circ}7$ and inequalities against the derivatives of a function of order $n$ (see further no. 16), we can demonstrate that in the case of a function of order $n(\geq 1)$ we have the same,

		$\displaystyle(x-a)^{x}f(x)(x)\rightarrow 0,\quad k=1,2,\cdots,n-1,$
		$\displaystyle(x-a)^{n}f_{g}^{(n)}(x)\rightarrow 0,\quad(x-a)^{n}f_{x}^{(n)}(x)\rightarrow 0,$

when $E^{\prime}\Rightarrow x\neq a$ tends towards $a$ A similar property occurs at the right end. $b$ of E. We must not forget that we assume E is closed, therefore bounded, therefore $f$ is bounded on E.

In the case $E=$ open interval $(a,b)$ , the properties are more precise. For / to be non-convex resp. convex of order $n>1$ it is necessary or it is sufficient that $f(k),k\geq n-1$ , exists in ( $a,b$ ) and be non-concave of order n - . k. Similarly, it is necessary and it suffices that $f^{(n)}$ exists and is either non-decreasing or increasing, except perhaps on a set that is at most countable (for $n=1$ see [19]). If $f^{(n+1)}$ exists, the condition $f^{(n+1)}\geq 0$ is necessary and sufficient for non-concavity eu $f^{(n+1)}>0$ is sufficient for the convexity of $f$ We also have in this case $nA_{n}[f;E]==A_{n-1}\left[f^{\prime};E\right]$ And $nV_{n}[f;E]=V_{n-1}\left[f^{\prime};E\right]$ , whatever the limits and total variations, finite or infinite.

According to L. Gasvary [19], the existence of a single,
$x_{1}<\xi<x_{2}$ for everything $x_{1},x_{2}\in(a,b)$ in the mean value formula $\left[x_{1},x_{2};f\right]=f^{\prime}(\xi)$ , is necessary and sufficient for the first-order convexity or concavity of the differentiable function $f$ in (a, b). A. Terracini [58] also notes that one has $\xi\geq\frac{x_{1}+x_{2}}{2}$ depending on $f^{\prime\prime}\cdot f^{\prime\prime\prime}\geq 0$ assuming the existence of these derivatives.

If $f$ is of order $n>2$ And $\Delta_{0}$ is the maximum of $|f|$ in the closed interval ( $a,b$ ), we have
(10) $\left|f^{\prime}\right|\leqq\overline{b-a},\left|f^{\prime}\sqrt{(x-a)(b-x)}\right|\leqq\mathrm{B}n\Delta_{0},\quad x\in(a+\lambda,b-\lambda)$
with

\begin{gathered}A=8(7+4\sqrt{3})/3<38,\quad B=2(7+4\sqrt{3})<28\\ \lambda=(b-a)\left(1-\cos\frac{\pi}{2n}\right)/\left(1+\cos\frac{\pi}{n}\right)\end{gathered}

Inequalities (10) are analogous to inequalities

|f|\leqq\frac{2n^{2}\Delta_{0}}{b-a},\quad\left|f^{\prime}\sqrt{(x-a)(b-x)}\right|\leqq n\Delta_{0},\quad x\in(a,b)

by A. Markofp [37] and S. Bernstein [7b], when $f$ is a polynomial of degree $n$ P.
Montel [39 b] noted, in the case $n=0$ that the integral $\int_{a}^{r}f(t)dt$ , of a non-concave or convex function of order $n$ , is non-concave resp convex of order $n+1$ Moreover, we can see that if $k$ is a positive integer, and $f$ summable and non-concave of order $n$ In $(a,b)(n\geq 1)$ , the integral of order $k$

\int_{a}^{x}(x-t)^{k-1}f(t)dt

is non-concave of order $n+k$ In $(a,b)$ .
If $\varphi(t)$ is a non-decreasing and bounded function in the interval ( $a,b$ ), the complete works of Stieltjes

f(x)=\int_{a}^{b}\mathrm{G}_{n}(x,t)d\varphi(t)

(11)

\mathrm{G}_{n}(x,t)=\frac{1}{n!}\left[\frac{|x-t|+x-t}{2}\right]^{n},

exists and represents a non-concave function of order n with a bounded name. d. $(a,b)$ Moreover, we can see that if $n>1$ ,

		$\displaystyle\qquad f(k)(x)=\int_{a}^{b}G_{n-k}(x,b)d\varphi(t),\quad k=1,2,\cdots,n-1$
		$\displaystyle\text{ et si }n\geqq 1,$
		$\displaystyle\qquad f_{g}^{(n)}(x)=\varphi(x-0)\cdots\varphi(a),\quad f_{d}^{(n)}(x)=\varphi(x+0)+\varphi(a)$
		$\displaystyle\text{ pour tout }x\in(a,b)\text{. }$
		$\displaystyle\text{ Réciproquement, toute fonction non-coñcave d'ordre }n(\geqq 1)$
		$\displaystyle\text{ ot à nème d. d. bornée dans }(a,b)\text{ peut se mettre, à un polynome }$
		$\displaystyle\text{ additif de degré }n\text{ près, sous la forme (11). Il suffi, de prendre, par }$
		$\displaystyle\text{ exemple, }\varphi(t)=f_{d}^{(n)}(t)\text{ et d'ajouter le polynome }$

\sum_{i=0}^{n}\frac{(x-a)^{i}}{i!}f^{(i)}(a).

W. Blaschke and G. Pick [9] demonstrated, in the case $n=1$ that representation by a Stieltjes integral is always possible. Indeed, the integral

\Phi(t)=\int_{c}^{t}(x-a)(b-x)df_{d}^{\prime}(x)

Or $c=\frac{a+b}{2}$ exists for $a<t<b$ and is a non-decreasing function, bounded in the open interval ( $a,b$ By advocating for $\Phi(a)$ a suitable value $\leqq\Phi(a+0)$ and for $\Phi(b)$ a suitable value $\geq$ □ ( $b-0$ ), we have the representation

f(x)=\frac{1}{b-a}\int_{a}^{b}H(x,b)d\Phi(b)

lawyer

\mathbf{H}(x,t)=\begin{cases}\frac{x-b}{b-t},&t\leq x\\ \frac{a-x}{t-a},&t\leq x\end{cases}

valid for all points $x$ inside the interval ( $a,b$ 12.
- We can also study derivatives defined differently.

To simplify, suppose $f$ defined within the bounded interval ( $a,b$ Suppose that at all $x\in\cdot(a,b)$ corresponds a set $e$ of pairs of points ( $x^{\prime},x^{\prime\prime}$ ), such as $x^{\prime}<x<x^{\prime\prime}$ and that all
kingship $\mathrm{V}_{x}^{1}$ contains at least one pair ( $x^{\prime},x^{\prime\prime}$ ). The max. of the min. of $2\left[x^{\prime},x,x^{\prime\prime};n\right]$ tend, for $\lambda\rightarrow 0$ , when the points $x^{\prime},x^{\prime\prime},\left(x^{\prime},x^{\prime\prime}\right)\in e$ remain in the interval ( $x-\lambda,x+\lambda$ ), towards the limits (proper or improper) $\overline{\mathbf{D}}^{2}f(x),\underline{\mathbf{D}}^{2}f(x)$ We can call $\bar{D}^{2}/(x)$ a generalized upper second derivative in $x$ And $\mathrm{D}^{2}f(x)$ a generalized lower second derivative in $x$ If, in every respect $x\in\cdot(a,b)$ we have defined the numbers $\overline{\mathrm{D}}^{2}f(x),\overline{\mathrm{D}}^{2}f(x)$ , we have a generalized upper second derivative resp lower second derivative in ( $a,b$ ). If $f$ is twice differentiable, obviously. $\overline{\mathrm{D}}^{2}f(x)=\bar{D}^{2}f(x)=f^{\prime\prime}(x)$ .

We then have the following property:
If $f$ is semi-continuous from above and if there exists an upper second derivative $\mathbb{D}^{2}f(x)$ such as

\bar{D}^{2}f(x)\geqq 0,\quad x\in\cdot(a,b)

the function $f$ is non-concave of order 1 in $(a,b)$ Indeed
, either $\mu=$ min $\bar{D}^{2}f(x)$ In $(a,b)$ . We have $\mu\geq 0$ . If $\mu>0$ , ownership results from ownership demonstrated in paragraph 8. For $\mu=0$ , the property results from the fact that then $f+\alpha x^{2}$ Or $\alpha>0$ is in the case $\mu>0$ The functions $f+\alpha x^{2},\alpha>0$ are therefore non-concave of order 1, the same is true for their limit $f$ if $\alpha\rightarrow 0$ (see further on) $\mathrm{n}^{\circ}17$ ).

In the particular case $x^{\prime}=x\cdots h,x^{\prime\prime}=x+h,0<h<\varepsilon$ , $\varepsilon>0,\mathrm{D}^{2}f(x),\mathrm{D}^{2}f(x)$ are, the usual generalized second derivatives and the previous property is due to S. Saks [51 a].

CHAPTER II

VARIOUS PROPERTIES OF ORDER FUNCTIONS n

13.
- •
  
  For a non-concave function of order 1 we have the classic inequality

f\left(\frac{p_{1}x_{1}+p_{2}x_{2}+\cdots+p_{m}x_{m}}{p_{1}+p_{2}+\cdots+p_{m}}\right)\leqq\frac{p_{1}f\left(x_{1}\right)+p_{2}f\left(x_{2}\right)+\cdots+p_{m}f\left(x_{m}\right)}{p_{1}+p_{2}+\cdots+p_{m}},

(12)

Or $p_{i}>0$ , given in a particular case by O. Hölder [22], and in the general case by JLWV Jensen [27]. If $f$ is convex, equality is only possible for $x_{1}=x_{2}=\ldots=x_{m}$ A.
del Chiaro [15] generalized this inequality with the following

f\left[\frac{\mid p\varphi dx}{\int pdx}\right]\leqq\frac{\int pf(\varphi)dx}{\int pdx},

(13)

the integrals being taken from $a$ has $b$ and where $p(x)$ is summable and almost everywhere $>0,\varphi(x)$ measurable and bounded in $(a,b),f(x)$ nonconcave of order 1 in $(m,\mathrm{M}),m=\min\varphi,\mathrm{M}=\max$ . If $f$ is convex and q is not almost everywhere a constant, we have the sign ¡ in (13). In particular cases inequality (13) has already been pointed out by JLWV Jensen [27] and G. Pólya [46 a].

All these inequalities result from the following principle: Either $f$ non-concave of order 1 on E and F. a family of functions $\varphi$ of one or more variables, defined on a set $\mathrm{E}^{*}$ so that: ${}^{10}\varphi$ is uniform and all its values belong to E, 20 if $\varphi_{1},\varphi_{2}\in\mathscr{I}$ we also have $k\varphi_{1}\in\mathfrak{J},\varphi_{1}+\varphi_{2}\in\mathscr{F},k$ being a real constant, $3^{0}$ if $\varphi\in\mathscr{I}$ we also have $f(\varphi)\in\mathscr{I},4^{0}1\in\mathscr{I}$ Let Ω[φ] then be a non-negative linear functional operation, defined for the family $\mathscr{F}^{\circ}$ , whose values belong to E and such that $\Omega[1]=1$ We have done:
$1^{\circ}\Omega[k\varphi]=k\Omega[\varphi]$ if $k$ is a constant,
$2^{\circ}\Omega\left[\varphi_{1}+\varphi_{2}\right]=\Omega\left[\varphi_{1}\right]+\Omega\left[\varphi_{2}\right]$ ,
$3^{\circ}\Omega[\varphi]\geqq 0$ if $\varphi\geqq 0$ Inequality

f(\Omega[\varphi])\leqq\Omega[f(\varphi)],

then results from (6) as noted by B. Jessen [28 a]. By particularizing the family sh and choosing the operation appropriately $\Omega$ EJ MeShane [38] obtained various inequalities and also studied the cases where the = sign occurs. In particular, the family can be formed by a certain class of functions of any number $m$ of variables and the operation $\Omega$ expressed by an integrate $m$ -uple.
HP Mulholland [40] demonstrates that, in order for one to have

f\left(\frac{\Sigma p_{i}x_{i}}{\Sigma p_{i}}\right)\leqq\mathrm{A}\frac{\Sigma p_{i}f\left(\mathrm{~B}x_{i}\right)}{\Sigma p_{i}},\quad p_{i}>0,

Given two positive constants A and B, it is necessary and sufficient that we can find a function $\varphi$ non-concave of order 1 such that

\varphi(x)\leqq f(x)\leqq\mathrm{A}\varphi(\mathrm{~B}x).

Let us consider the quasi-arithmetic mean

\Re_{\varphi}=\varphi^{-1}\left(\frac{\Sigma p_{i}\varphi\left(x_{i}\right)}{\Sigma p_{i}}\right),\quad\text{ ou }\quad\Re_{\varphi}=\varphi^{-1}\left(\int_{0}^{1}\varphi(g)dx\right),

$g$ being a suitable function of $x$ And $\varphi^{-1}$ the inverse function of $\varphi$ . B. Jessen [28 a, 28 b] demonstrates that if $\varphi$ , $\psi$ are continuous $\varphi$ monotonous, $\psi$ growing inequality $\mathcal{M}_{\varphi}\leqq\mathcal{M}_{\psi}$ for all $x_{i},p_{i}>0$ or the functions $g$ This amounts to Jensen's inequality. Indeed, for this to be the case, it is necessary and sufficient that $\psi\left(\varphi^{-1}\right)$ either nonconcave of order 1. K. Knopp [32 a] remarks, moreover, that, for this to be so, it is necessary and sufficient that

\varphi^{-1}\left(\frac{\varphi(x)+\varphi(y)}{2}\right)\leqq\psi^{-1}\left(\frac{\psi(x)+\psi(y)}{2}\right)

for all $x,y$ in the interval of definition of the functions.
14. - Inequalities upper limiting the second number of (12) or (13) have been established under certain restrictive hypotheses made on the $x_{i}$ or on the functions $p,\varphi$ .

⁰⁰footnotetext: Tibere Popovigiu.

Let f be a continuous, non-concave function of order 1 in the interval formed $(0,1)\left({}^{1}\right)$ Let's ask.

\mathrm{A}=\int_{0}^{1}\varphi dx,\quad\mathrm{~A}_{j}=\int_{0}^{1}f(\varphi)dx

where, to simplify, we can assume p is continuous in $(0,1)$ , such as $0\leqq\alpha\leqq\phi\leqq\beta\leqq 1,\alpha<\beta$ Jensen's inequality is written alora $\mathrm{B}_{f}=\Lambda_{f}-f(A)\geqq 0$ Furthermore, if $\varphi$ is non-concave of order $0,1,\ldots,n$ we have [47 1 ],

\Lambda_{l}\leq\int_{0}^{1}fl\alpha+j(j+1)\left[\left(\mathrm{A}-\frac{jx+\beta}{j+1}\right)+\left(\frac{(j-1)\alpha+\beta}{j}-\mathrm{A}\right)x\right]x^{j-1},dx

(14)

\frac{j\alpha+\beta}{j+1}\leqq\Lambda\leqq\frac{(j-1)\alpha+\beta}{i},\quad 2\leqq i\leqq n

and.
(15) $A_{1}\leq\frac{\beta+n\alpha-(n+1)A}{\beta-\alpha}f(\alpha)+\frac{(n+1)(A-\alpha)}{n(\beta-\alpha)\sqrt[n]{\beta-\alpha}}\int_{\alpha}^{\beta}\frac{f(x)dx}{\sqrt[n]{(x-\alpha)^{n-1}}}$ if $a\subseteq A\leqq\frac{n\alpha+\beta}{n+1}$ .

Furthermore, if the function $f$ is convex, the equality in (14) is only possible if φ is equal to the polynomial that appears as the argument of $f$ in the second member and in (15) only if

\psi=\alpha+(\beta-\alpha)\left[\frac{|x-\lambda|+x-\lambda}{2(1-\lambda)}\right]^{n},\quad\lambda=\frac{\beta+n\alpha-(n+1)\mathrm{A}}{\beta-\alpha}.

These results lead to the upper limit of $\mathbf{B}_{f}$ According to K. Knopp [32 b], if p is non-decreasing,

\mathrm{B}_{1}\leqq\max_{(\alpha,\beta)}\left[\frac{(\beta-x)f(\alpha)+(x-\alpha)f(\beta)}{\beta-\alpha}-f(x)\right].

The maximum of the right-hand side is reached for only one value $x_{1}$ of $x$ , if $f$ is convex. The maximum of $\mathrm{B}_{f}$ is then only reached by discontinuous functions, for example by

\varphi=\begin{cases}\alpha,&0\leqq x\leqq\frac{\beta-x_{1}}{\beta-\alpha},\\ \beta,&\frac{\beta-x_{1}}{\beta-\alpha}\leq x\leq 1.\end{cases}

(1) We can easily move to a closed interval ( $a,b$ ) any.

When e is continuous, one-decreasing and non-concave of order 1, we have [47, 4 ],

\left.B_{1}\leqq\max_{\left(\alpha,\frac{\alpha+\beta}{2}\right)}f(\alpha)+\frac{2(x-\alpha)}{\beta-\alpha}\int_{\alpha}^{\beta}[f(t)-f(\alpha)]dt-f(x)\right\}

equality is only possible, if f is convex, for a single value $x_{3}$ of $x$ and for the function

\varphi=\alpha+(\beta-\alpha)\frac{|x-\lambda|+x-\lambda}{2(1-\lambda)},\quad\lambda=\frac{\beta+\alpha-2x_{0}}{\beta-\alpha}.

In the demonstration we gave of these results [47r], we further assumed that $f$ has finite derivatives at 0 and 1. It can easily be seen that this restriction is not essential.
J. Favard [16] demonstrated, among other things, that

A_{I}\leqq\int_{0}^{1}f(2xA)dx

if $\varphi,0\leqq m\leqq\varphi\leqq\mathrm{M}$ , is non-convex of order 1 in $(0,1)$ And $j$ non-concave (integrable into $\left(0,M_{1}\right),M_{1}=\max(M,2A)$ . If $f$ Furthermore, since it is convex, equality is only possible for functions
$\varphi=\mu x,\quad\mu(1-x),\quad\mu[(1-2\lambda)x+\lambda-|x-\lambda|],\quad\mu>0,\quad 0<\lambda<1$ 15.
- We sought to determine inequalities of the form
(16) $\sum_{i=1}^{m}p_{i}f\left(x_{i}\right)\geqq 0,\quad x_{1}<x_{2}<\cdots<x_{m},\quad p_{i}\neq 0,\quad i=1,2,\ldots,m$ valid
for all non-concave functions of order n. The necessary and sufficient condition is that we have [47 h, 47 [],

	$\displaystyle\sum_{i=1}^{m}p_{i}x_{i}^{k}=0,\quad k=0,1,\ldots,n$		(17)
	$\displaystyle\sum_{i=1}^{r}p_{i}\left(x_{i}-x_{r+1}\right)\left(x_{i}-x_{r+2}\right)\cdots\left(x_{i}-x_{r+n}\right)\leqq 0$
	$\displaystyle r=1,2,\ldots,m-n-1$

We assume, of course, that this is a function defined on the points $x_{i}$ When the function is defined in an interval containing the points $x_{i}$ The previous conditions are no longer
necessary for $n>1$ In this case, the necessary and sufficient conditions are (17) and

-\sum_{r=1}^{r}p_{i}\left(x_{i}-x\right)^{n}=\sum_{\begin{subarray}{c}i=r+1\\ r=1,2,\ldots,m-n-1\end{subarray}}^{m}p_{i}\left(x_{i}-x\right)^{n}\geq 0,\quad x\in\left(x_{r},x_{r+1}\right),

When the function is convex of order $n$ the sign ¿ takes place in (16).

Consider $r$ points $x_{1}<x_{2}<\ldots<x_{r}$ And $r$ positive numbers $\lambda_{1},\lambda_{3},\ldots,\lambda_{r}$ Among all polynomials of degree $m$ of the shape $P(x)=x^{n}+\ldots$ There is one and only one $\mathrm{P}_{m}(x)$ making minimum Pexpression $\sum_{i=1}^{r}\lambda_{i}\left[\mathrm{P}\left(x_{i}\right)\right]^{3}$ We thus have a sequence of polynomials.

1=P_{0},P_{1},\ldots,P_{r-1},\quad P_{r}=\prod_{i=1}^{r}\left(x-x_{i}\right)

which are orthogonal, in the sense that

\sum_{i=1}^{\sum_{i=1}P_{\alpha}\left(x_{i}\right)P_{\beta}\left(x_{i}\right)=0,\quad\alpha\neq\beta.}

The zeros of the polynomial $\mathrm{P}_{m}(m<r)$ are all real, distinct, and within the interval ( $x_{1},x_{r}$ Furthermore, the zeros of the polynomial $P_{s,s}\leq m$ are separated by zeros $y_{1}\leq y_{2}\leq\ldots\leq y_{m}$ of $P_{m}$ , $m\leqq r$ that is to say, the following

\mathrm{P}_{s}\left(y_{1}\right),\mathrm{P}_{s}\left(y_{2}\right),\quad\ldots,\mathrm{P}_{s}\left(y_{m}\right)

present $s$ variations [47 b], which greatly clarifies the distribution of points $x_{i},y_{i}$ .

The zeros $y_{i}$ of $\mathrm{P}_{m}$ are determined by the system

\sum_{i=1}^{m}\mu_{i}y_{i}^{s}=\sum_{i=1}^{r}\lambda_{i}x_{i}^{s},\quad s=0,1,\ldots,2m-1,

which also gave the positive numbers $\mu_{i},i=1,2,\ldots,m$ , which we call the weights of the polynomial $\mathrm{P}_{m}$ . THE $\lambda_{i}$ are the weights of $\mathrm{P}_{r}$ .

Any function defined on the points $x_{i},y_{i}$ and non-concave, odd order $n=2m-1$ , checks the inequality

\sum_{i=1}^{m}1n_{i}f\left(y_{i}\right)\leqq\sum_{i=1}^{n}\lambda_{i}/\left(x_{i}\right),\quad\left(m\geq 1,\quad r>\frac{n+1}{2}\right).

For $r=m+1$ The inequality is equivalent to the inequality of definition. If, in addition, the function is convex, we have the sign $<[47\mathrm{i}]$ .

Let's now $z_{1}<z_{2}<\ldots<z_{m}$ the zeros, all real and distinct, of the polynomial $\mathrm{P}_{n}+\rho\mathrm{P}_{m+1}$ We have the system

\sum_{i=1}^{m}v_{i}z_{i}^{s}=\sum_{i=1}^{r}\lambda_{i}x_{i}^{s},\quad s=0,1,\cdots,2m-2

which also determines the positive weights $\nu_{i}$ of the polynomial $\mathrm{P}_{m}+\rho\mathrm{P}_{m+1}$ Any
non-concave function of even order $n=2m-2$ , defined on the points $x_{i},z_{i}$ checks the inequality

\sum_{i=1}^{m}v_{i}f\left(z_{i}\right)\equiv\sum_{i=1}^{r}\lambda_{i}f\left(x_{i}\right),\quad\left(m\geq 1,\quad r\geq\frac{n+2}{2}\right)

provided that the $\rho$ be chosen in such a way that $y_{1}\leqq x_{1}$ , which for $r=m$ returns to the inequality of definition. When the function is convex the sign ¡ takes place [47 1].

For $m=1$ we can also see that, if $f$ is continuous and increasing, or decreasing, there is only one $z_{1}$ such as

f\left(z_{1}\right)=\frac{\sum_{i=1}^{r}\lambda_{i}f\left(x_{i}\right)}{\sum_{i=1}^{r}\lambda_{i}},\quad x_{1},\leq x_{2}\leq\cdots\leq x_{r},

and, if $x_{1}<x_{r}$ we have $x_{1}<z_{1}<x_{r}[210]$ In
the case $n$ anything, we can do $r\rightarrow\infty$ and thus obtain inequalities analogous to that of JLWV Jensen [see 47 1].
16. - Inequality (16) can also be written in the form

\sum_{i=1}^{m}f\left(x_{i}\right)\geq\sum_{i=1}^{m}f\left(y_{i}\right),

(18)

$x_{1}\leqq x_{2}\leqq\ldots\leqq x_{m},y_{1}\leqq y_{2}\leqq\ldots\leqq y_{m}$ , if the mutual ratios of the coefficients $p_{i}$ are rational.

For this inequality to hold true for any non-decreasing function, it is necessary and sufficient that $x_{i}\geq y_{i},i=1,2,\ldots m$ , the cases of equality being immediate.
GH Hardy, JL Liptlewood and G. Pólya [21 a] and J. Kaba-

MATA [31] examined the inequality (18) for a nonconcave conjunction of order 1. The necessary and sufficient condition sought is then

\begin{gathered}x_{1}+x_{2}+\cdots+x_{i}\leq y_{1}+y_{2}+\cdots+y_{i},\quad i=1,2,\ldots,m-1,\\ x_{1}+x_{2}+\cdots+x_{m}=y_{1}+y_{2}+\cdots+y_{m}.\end{gathered}

This condition is equivalent to the following [21 b]: We can find $m^{2}$ non-negative numbers $p_{ij}$ such as

	$\displaystyle y_{i}=$	$\displaystyle p_{i1}x_{1}+p_{i2}x_{2}+\cdots+p_{im}x_{m,}\quad i=1,2,\cdots,m$
		$\displaystyle\sum_{i=1}^{m}p_{ij}=\sum_{j=1}^{m}p_{ij}=1,\quad i,j=1,2,\cdots,m$

This category includes the K. Tons inequality [59]

\frac{1}{m}\sum_{i=1}^{m}f\left(x_{i}\right)\geq\frac{1}{m-1}\sum_{i=1}^{m-1}f\left(y_{i}\right)

valid for first-order non-concave mowing functions, the $y_{i}$ being the zeros of the derivative of the polynomial $\prod_{i=1}^{m}\left(x-x_{i}\right)$ equality is only possible for a convex function if $x_{1}=x_{2}=\ldots=x_{m}$ For functions of the form $x^{p}(p\geq 1$ Or $p<0)$ , this inequality has already been studied by HE Bray [12] and S. Kakeya [30 c].

Mr. Pethovitch's inequality [44]

\sum_{i=1}^{m}f\left(x_{i}\right)\leqq(m-1)f(0)+f\left(\sum_{i=1}^{m}x_{i}\right),\quad x_{i}\geq 0,

or the more general inequality

\sum_{i=1}^{9}p_{i}f\left(x_{i}\right)\leq\left(p_{i}-1\right)f(0)+1\left(\sum_{i=01}^{9}p_{i}x_{i}\right),\quad x_{i}\geq 0,\quad p_{i}>0

Or $f$ is non-concave of order 1 in $(0,+\infty)$ closed on the left, can. be linked to inequalities (16) and (18).

So that we may have, $\varphi,\psi$ , being non-decreasing.

\int_{a}^{b}f(\varphi)dx\leqq\int_{a}^{b}f(\psi)dx

(19)

for any function $f$ continuous and non-concave of order 1, it is necessary that eb it suffices that

\left.\int_{\xi}^{b}\varphi dx\leqq\int_{\xi}^{b}\psi dx,\quad a\leqq\zeta\leqq b\quad\text{ (pour }\xi=a,\text{ le signe est }=\right)

or, in another form equivalent to this one,

\left.\int_{a}^{b}|\varphi-\lambda|dx\leqq\int_{a}^{b}|\psi-\lambda|dx,\quad\text{ (pour }\lambda=0,\text{ le signe est }=\right)

whatever the constant $\lambda$ [21 b].
All these results are easily obtained by noting that, for an inequality of the indicated form to be true for any non-concave function of order $n$ , it is necessary and sufficient that it be true for a polynomial of degree $n$ and for functions of the form $(|x-\lambda|+x-\lambda)^{n}$ , $\lambda$ being a constant.

As an application of (19), we can consider the inequality

\frac{1}{2h}\int_{x\rightarrow h}^{x+h}f(t)dt\leqq\frac{f(x+h)+f(x-h)}{2},\quad x+h,x-h\in\cdot(a,b).

which is satisfied by any non-concave function of order 1 in ( $a,b$ Conversely, any summable and semi-continuous function above, which satisfies this inequality, regardless of $x$ ot $h$ possible, is non-concave of order 1 in $(a,b)$ T.
Rado [48b] generalized this inequality as follows: the inequality

	$\displaystyle{\left[\frac{1}{2h}\int_{x-h}^{x+h}[f(t)]^{\alpha}dt\right]^{\frac{1}{\alpha}}\leqq\left[\frac{f(x+h)^{\beta}+f(x-h)^{\beta}}{2}\right]^{\frac{1}{\beta}}}$		(20)
	$\displaystyle x+h,x-h\in\cdot(a,b)$

is satisfied by any continuous, positive, and non-concave function of order 1 in $(a,b)$ if, and only if $\alpha\leq-2,\beta\geq 0$ , Or $-2\leqq\alpha\leqq-\frac{1}{2},3\beta\geqq\alpha+2$ , Or $-\frac{1}{2}\leqq\alpha\leqq 1,\beta\supseteq\alpha\log 2/\log(\alpha+1)$ or, finally, $\alpha\geq 1,3\beta\geq\alpha+2$ . If $3\beta-\alpha-2\leq 0$ and if the function is continuous and positive $f$ satisfies inequality (20), it is nonconcave of order 1 in $(a,b)$ It follows that continuous, positive, and non-concave functions of order 1 are characterized by inequality (2), if $3\beta=\alpha+2et-2\leq\alpha\leq-\frac{1}{2}$ Or $\alpha\geq 1$ 17.
— Inequalities can also be established between the values of the function $f$ and its derivatives up to a certain order. To simplify, suppose $f$ defined in an interval containing the points $x_{1}<x_{2}<\cdots<x_{m}$ . For inequality (21) $\sum_{i=1}^{m}\sum_{i=0}^{k_{i}}p_{ij}(i)\left(x_{i}\right)\geq 0,\quad\sum_{i=0}^{k_{i}}\left|p_{ij}\right|\neq 0,\quad i=1,2,\ldots,m_{i}$ ,
oin $0\leqq k_{i}\leqq n$ ot $f^{(n)}$ denotes one of the derivatives $f_{g}(n),f_{d}^{(n)}$ (not necessarily the same for everyone) $x_{1}$ ), either verified for Touba, a non-concave function of order n, it is necessary that it suffices that

		$\displaystyle\sum_{i=1=1}^{m}\sum_{j=0}^{k_{i}}\operatorname{pys}_{i}(s-1)\cdots(s-j+1)x_{i}^{s-j}=0,\quad s=0,1,\ldots,n,$
		$\displaystyle\sum_{i=r+1j=0}^{m}p_{ij}n(n-1)\cdots(n-j+1)\left(x_{i}-x\right)^{x_{n}-1}\geq 0,\quad x\in\left(x_{r},x_{r+1}\right)\text{, }$
		$\displaystyle r=1,2,\ldots,m-1.$

In particular, in the case $\sum_{i=1}^{m}\left(k_{i}+1\right)=n+2$ Inequality (21) is simply the limit of the defining inequality (1) when the points $x_{i}$ tend towards each other in groups of $k_{i}+1,k_{a}+1$ , …, sem +1 points. This inequality can be written in the form

\left[\frac{x_{1},x_{1},\ldots,x_{1}}{k_{1}+1},x_{2},x_{2},\ldots,x_{2},\ldots,x_{m,}^{x_{m}+1},\ldots,x_{m};n\right]\geq 0

and the first member is a quotient of two determinants that can be easily obtained from the determinant of order $n+1$ by repeated application of LjHospital's rule. In particular, for $m=2,k_{1}=k$ we obtain

	$\displaystyle(-1)^{n-k+1}$	$\displaystyle{\left[\sum_{i=0}^{k}\binom{k}{i}(n-\cdots,\cdots-a)^{i}f(i)(a)-\right.}$
		$\displaystyle\left.-\sum_{i=0}^{n-k}(-1)^{f}(n-k)(n-i)!(b-a)^{if(i)}(b)\right]\geq 0,\quad a<b.$

If the function $f$ is convex of order $n$ In all these inequalities, the symbol ¿ appears.
18. A function can possess several convexity properties simultaneously. In particular, functions that are non-concave of order $0,1,\ldots,n$ are also called ( $n+1$ - monotonic and appear in various problems of Mathematical Analysis. Non-concave functions of any non-negative ontive order are also called completely monotonic functions. We will say a word about these functions in the next chapter.

There exist functions defined on a finite set, enjoying
several convexity properties given in advance, provided that, if the order property $n$ is polynomial, any property of order $>n$ or also of polynomiality. Moreover, a polynomial function of order $n$ must necessarily be convex, polynomial or concave of order $n-1$ We can also find functions (polynomials) enjoying a finite number of convexity or concavity properties chosen arbitrarily within an interval $(a,b)$ [47 d]

The family of order functions $n$ is invariant under a linear transformation. If $f,g$ are functions enjoying the same convexity properties, the functions cf, $f+g$ also enjoy the same properties if $c$ is a positive constant, convexity and polynomiality being considered special cases of non-concavity. We can also specify the convexity characteristics of the product $fg$ and the function of function $f(g)$ For example, if $f$ is non-concave of order - $1,0,1,\ldots,n$ And $g$ non-concave order - $1,0,1,\ldots,n$ or non-convex of order - $1,0,1,\ldots,n$ the product $fg$ is also non-concave or non-convex of order -1.0, $1,\ldots,n$ . If $f,g$ are $n$ -moon-twisted, $f(g)$ is also $n$ -times monotonic,… etc. Such properties have already been established by JLWV Jensen [27] for monotonic functions and first-order functions.

The limit of a convergent sequence $\left|f_{m}\right|$ of functions enjoying the same convexity properties, still enjoys the same properties. According to P. Montel [39b] if the $f_{m}$ are non-concave of order 1 and also bounded on a subset completely interior to E, $\overline{\text{ im }}f_{m}$ and max $f_{m}$ are also non-concave of order 1 on this subset. These properties remain true for functions of order -1 or 0 and also for functions that possess two or three properties of order -1, 0 and 1, but are obviously not true for functions of order $>1$ To see this, one only needs to consider the family of two functions $x$ , - $x$ which are of order $n>1$ , but their maximum function $|x|$ is not in order $n>1$ in an interval containing the origin. G. Valiron [61 a] clarified the previous results by noting that if $f(x;t)$ is continuous in $x$ And $t$ For $a\leqq x\leqq b,\alpha\leqq t\leqq\beta$ and convex of order 1 in $x$ for everything $t$ the max function $f(x;t)$ on $\mathbf{E}_{t}\subset(\alpha,\beta)$ is also convex of order 1 in $(a,b)$ The example
$f(x;t)=\frac{\sqrt{16-t^{2}}-\sqrt{16-(2x-t)^{2}}}{2},\quad-1<x<1,\quad-1\leqq t\leqq 1$ This
shows us that this property cannot be extended to the case $n>1$ On the contrary, as G. Valiron [ $\mathbf{61}$ al] also notes this in the case $n=1$ the complete $\int_{x}^{\beta}f(x;t)dt$ is non-concave or convex of order $n$ if $f(x;i)$ is non-concave resp convex of order n for Lout $t$ Here, it suffices that the integral exists in Lebesgue's sense, regardless of $x$ and we can, of course, replace ( $a,b$ ) by an arbitrary set E, and ( $x,\beta$ ) by an arbitrary measurable set.

Any family of non-concave functions of order 1 and equally bounded on E is also equally continuous, and therefore normal on any section c. E.M. Nicolesco [41] proved that the family is normal provided that the functions are equally bounded above, and deduced that any monotonic sequence of non-concave functions of order 1 that converges at a point $x_{0}\in\cdot E$ converges uniformly on any section c. E.
19. - P. Montea [39 b] and G. Valiron [61 a] have shown that for that $\log/$ either non-concave or convex of order 1, it is necessary and sufficient that $e^{xy}f$ either non-concave or convex of order 1 for all values of the constant $\alpha$ .

Either $F(t)$ a function increasing in ( $-\infty,+\infty$ ). If the function $F(f+P)$ is non-concave or convex of order $n(\geq 1)$ on E, for any polynomial P of degree $n$ without a constant term, the function $f$ is non-concave or convex of order $n$ on E. Indeed, if we determine the polynomial P such that

f\left(x_{i}\right)+P\left(x_{i}\right)=p,\quad i=1,2,\ldots,n+1,

We have

f\left(x_{n+2}\right)+P\left(x_{n+2}\right)=\rho+\frac{U\left(x_{1},x_{2},\ldots,x_{n+2};f\right)}{V\left(x_{1},x_{2},\ldots,x_{n+1}\right)}

and inequality

\left[x_{1},x_{2},\ldots,x_{n+2};F\right]\geqq 0\text{ resp. }>0,

gives us

\mathrm{F}\left(\int\left(x_{n+2}\right)+\mathrm{P}\left(x_{n+2}\right)\right)\geqq\mathrm{resp}.>\mathrm{F}(\rho),

Hence the property. For $n=0$ The property is commonplace. If $F(f)$ is non-decreasing or increasing, $f$ obviously non-decreasing resp increasing on E.

But there are much more comprehensive results. Suppose $F(t)$ continues in ( $-\infty,++\infty$ ), and either $f$ a continuous function in the interval ( $a,b$ Following a line of reasoning by S. Saks [51c], it can be shown that, if $\mathrm{F}(f+\alpha x+\beta)$ is non-concave of order 1, regardless of the constants $\alpha,\beta$ , SO : $1^{\circ}\mathrm{F}$ is non-concave of order 1; $2^{\circ}$ or $F$ is, constant, or $f$ is linear, on F is non-decreasing ol $f$ is non-concave of order 1, or finally F is non-increasing and $f$ is non-convex of order 1. If $n>1$ The questions become much simpler. Let's assume, to simplify things further, that $F$ And $f$ are ( $n+1$ ) differentiable times. So, if $\mathrm{F}(/-\mathrm{F})$ is non-concave of order n, for any polynomial P of degree $n$ The function F is necessarily linear. That is, in effect, $\xi$ a value of $t,x_{0}$ a value of $x$ And $\alpha$ any number. We can determine the polynomial P such that we have

	$\displaystyle f\left(x_{0}\right)+\mathrm{P}\left(x_{0}\right)=\xi,$	$\displaystyle f^{(k)}\left(x_{0}\right)+\mathrm{P}(k)\left(x_{0}\right)=1,\quad k=1,2,\ldots,n-1$
		$\displaystyle f^{(n)}\left(x_{0}\right)+\mathrm{P}(n)\left(x_{0}\right)=\alpha.$

The formula for the derivative $(n+1)^{\text{àme d'une fonetion de fonetion }}$ then gives us, for $x=x_{01}$

\frac{d^{n+1}}{dx^{n+1}}\Gamma(f+P)=(n+1)\alpha F^{\prime\prime}(\xi)+A\geqq 0,

being a number independent of $\alpha$ We deduce from this, $\Gamma^{\prime\prime}(\xi)=0$ , done $F$ is linear. Moreover, we can see that, furthermore, or bion $F$ is constant, or F is non-decreasing and f is non-concavo of order $n$ or $F$ is non-increasing and / non-convex of order n.
P. Montel [39 b] and G. Valiron [61 a] also demonstrated that, if log / is non-concave resp convex of order 1 in $\log x$ the function $\log\int_{0}^{x}/(x)dx$ is also non-concave resp. convex of order 1 in $\log x$ 20.
– According to J. Pál [43], every continuous non-convex function of order 1 in a closed interval can be approximated indefinitely by non-convex polynomials of order 1. Every function $f$ , continues in the closed interval $(a,b)$ is the limit of a sequence of polynomials, enjoying the same convexity properties as f
and converges uniformly throughout the interval [47d]. Ceoi is realized by the S. Bernstern polynomials [7a].

\mathrm{P}_{n}=\mathrm{P}_{n}(x;f)=\frac{1}{(b-a)^{n}}\sum_{i=0}^{n}\binom{n}{i}f\left(a+i\frac{b-a}{n}\right)(x-a)^{i}(b-x)^{n-i}.

Furthermore, any continuous and non-concave function of order $n$ in (a, b) is the limit of a uniformly convergent sequence of polynomials of order $n$ In $(-\infty,+\infty)[470]$ .

There $k$ one terminal of $P_{m}$ East $\leqq$ the kth terminal of $f$ For $k=0,1$ and is generally the, like the boundary of $f$ For $k>1$ Similarly, the total heme variation of $\mathrm{P}_{n}$ East $\leq$ that of $f$ For $k=0,1$ and is generally that of / for $k>1$ [47d].

Moreover, we have

\left|f(x)-P_{n}(x;f)\right|\leqq\frac{3}{2}m\left(\frac{b-a}{\sqrt{n}}\right),\quad x\in(a,b),

$\omega(\delta)$ being the oscillation modulus of $/[47\mathrm{~d}]$ Any
continuous function of order $n$ In $(a,b)$ is the limit of a uniformly convergent sequence of elementary functions (see no. 6) of order $n$ [47 c]. Let us suppose, first of all, that $f$ garlic a continuous derivative of order $n-1$ in the closed interval $(a,b)$ This is then a non-concave function of order 1. A polygonal line can be found. $y=\varphi(x)$ inscribed in the curve $y=f^{(n-1)}(x)$ , so that we have

\left|f^{(n-1)}-\varphi\right|<\frac{(n-1)!\epsilon}{(b-a)^{n-1}},\quad x\in(a,b),

& ¿ q arbitrarily given in advance. φ is an elementary function of order 1. The elementary function of order n

\varphi^{*}(x)=\int_{a}^{b}\mathrm{C}_{n-2}(x;t)\varphi(t)dt+\sum_{i=1}^{n-2}\frac{(x-a)^{i}}{i!}f^{(i)}(a),

then verifies the inequality.

\left|f(x)\cdots\varphi^{*}(x)\right|<\varepsilon,\quad x\in(a,b).

In the general case, it suffices to note that one can first find a function $f_{1}$ having a ( $n-1$ )th continuous derivative such that max $\left|f-f_{1}\right|$ can be as small as you want.

Let us note, moreover, as L. Galvani does [19] in the
case $n=1$ , that for any non-concave function of order $n$ In ( $a,b$ ) we have a decomposition of the form

f(x)=f^{*}(x)+x_{i=1}^{\infty}c_{i}\left[\frac{\left|x-x_{i}\right|+x-x_{i}}{2}\right]^{n},\quad x\in\cdot(a,b),

Or $x_{i}\in(a,b),c_{i}=0,\sum_{i=1}^{\infty}c_{i}<+\infty$ And $f$ is non-concave of order $n$ having a derivative of order $n$ continues in the open interval $(a,b)$ Lea $x_{i}$ are the points of discontinuity of the $n^{\text{ome }}$ derived from $f$ eb

c_{i}=\frac{1}{n!}\left[f_{d}^{(n)}\left(x_{i}\right)-f_{g}^{(n)}\left(x_{i}\right)\right].

21.
- •
  
  The Chebyshev polynomial, or the polynomial of best approximation of degree $n,\mathrm{~T}_{n}$ of a continuous function $f$ , defined on a closed set E, is the polynomial that minimizes the expression max $|f-\mathrm{P}|$ on E, where P ranges over the set of polynomials of degree $n$ The polynomial $\mathrm{T}_{n}$ is characterized completely by the fact that $f-\mathrm{T}_{n}$ reached the values $\pm\max\left|/-\mathrm{T}_{n}\right|$ in at least $n+2$ consecutive points with alternating signs, as demonstrated by E. Borel [11].

If $f$ is, moreover, non-concave of order $n$ , not reducing to a polynomial of degree $n$ one can only find $n+2$ consecutive points where

f-\mathrm{T}_{n}=\pm\max\left|f-\mathrm{T}_{n}\right|

with alternating signs. In other words, if $f$ is non-concave of order $n$ , without being polynomial of order $n$ polynomials $\mathrm{T}_{n},\mathrm{T}_{n+1}$ are distinct, therefore $\mathrm{T}_{n+1}$ is indeed of degree $n+1$ .

Conversely, if $f$ is a continuous function in the closed interval $(a,b)$ and if, for any closed interval $(c,d)\subset(a,b)$ , the Chebyshev polynomial of degree $n+1$ of $f$ In $(c,d)$ is indeed of degree $n+1$ , the function $f$ is convex or concave of order $n$ In $(a,b)$ $[47]$ .

It can also be noted that, if $f$ is of order $n$ the maximum value $\left|f-T_{n}\right|$ is necessarily reached at the extremities $a$ And $b$ Moreover, we have

\left[f(a)-T_{n}(a)\right]\left[f(b)-T_{n}(b)\right]\leq 0\quad\text{ou}\quad\geq 0

depending on $n$ is even or odd.

Conversely, if $f$ is continuous in the closed interval $(a,b)$ and if, for any closed interval $(c,d)\subset(a,b)$ the polynomial $T_{n}$ of $f$ In $(c,d)$ checks the equalities

\left|f(c)-\mathrm{T}_{n}(c)\right|=\left|f(d)-\mathrm{T}_{n}(d)\right|=\max_{(c,d)}\left|f-\mathrm{T}_{n}\right|,

the function $f$ is of order $n$ In $(a,b)$ $[47]$ .

22. Let us consider the integral

I(\varphi)=\int_{a}^{b}\varphi(x)dx

If $f$ is measurable and bounded within the closed interval ( $a,b$ ), the expression $\mathrm{I}(|f-\mathrm{P}|)$ has a minimum when P ranges over the set of polynomials of degree $n$ This minimum is attained by at least one polynomial. $\mathrm{P}_{n}$ , which is unique if $f$ is continuous, as demonstrated by D. Jackson [26]. In particular, for $f=x^{n+1}$ , A. Korkine, G. Zolotarefe [31 bis] and M. Fujewara [18] have shown that
(22) $x^{n+1}-\mathrm{P}_{n}=\frac{(b-a)^{n+1}\sin(n+2)\theta}{2^{2n+2}}+\quad 0=\arccos\frac{2x-a-b}{\sin\theta};\quad$ a done $(n+2)\left(x^{n+1}-P_{n}\right)$ is the derivative of the polynomial

\frac{(b-a)^{n+2}}{2^{2n+3}}\cos(n+2)0

who stands out, among all those of the form $x^{n+1}+\ldots$ , the least possible number of zeros in the interval ( $a,b$ ). The zeros of polynomial (22) are

t_{k}^{(n)}=\frac{a+b}{2}+\frac{b-a}{2}\cos\frac{k}{n+2},\quad k=1,2,\ldots,n+1

We can therefore say that the minimum of $I\left(\left|x^{n+1}-P\right|\right)$ is, realized by the Lagrange polynomial $\mathrm{P}\left(t_{1}^{(n)},t_{2}^{(n)},\ldots,t_{n+1}^{(n)};x^{n+1}\mid x\right)$ .

Either $n=0$ And $f$ a monotonic (non-constant) function that can be assumed to be non-decreasing. The problem then becomes determining the minimum of $I(|f-\lambda|)$ as a function of $\lambda$ The minimum is still reached by $\mathrm{P}\left(t_{1}^{(0)};f\mid x\right)=f\left(\frac{a+b}{2}\right)$ which results from inequality

(2x-a-b)f(x)\geqq 2\int_{\frac{a+b}{2}}^{x}f(t)dt

equivalent to inequality $\left[x,x,\frac{a+b}{2};F\right]\geq 0$ , verified by the
function $\mathrm{F}(x)=\int_{a}^{x}f(t)dt$ , non-concave of order 1. H. Steinhaus [55] demonstrated that, if $n=1$ the minimum is provided by the polynomial $\mathrm{P}\left(t_{1}^{(1)},t_{2}^{(1)};f\mid x\right)$ if $f$ is of order 1. Finally, V. Hruska [24] showed that, for $n$ for any polynomial, the minimum is given by the polynomial $\mathrm{P}\left(t_{1}^{(n)},t_{2}^{(n)},\ldots,t_{n+1}^{(n)};f(x)\right.$ when $f^{(n+1)}$ exists and is $\geqq 0$ It is easy to deduce that the result remains valid if $f$ is only of order $n$ .

CHAPTER III

GENERALIZATIONS OF ORDER FUNCTIONS $n$

23.
- •
  
  It appears that O. Stolz [57] was the first to introduce convex functions, demonstrating that for the existence of left-hand and right-hand derivatives, ' $f_{g}{}^{\prime}$ And $f_{d}{}^{\prime}$ , at every point of an open interval ( $a,b$ ) where the continuous function $f$ is defined, it is sufficient that at all $x$ corresponds to a $n>0$ such as one has

f(x+h)+f(x-h)-2f(x)\geqq(\mathrm{ou}\leqq)0,

(23)

For $|h|<n$ This author demonstrates that (23) entails inequality

\left[x_{1},x_{2},x_{3};f\right]\geqq(\mathrm{ou}\leqq)0,\quad a\leqq x_{1}<x_{2}<x_{3}\leqq b.

But it was JLWV Jensen [27] who first studied systematically the functions satisfying inequality (23).

Let us consider the differences in order $n$ ,

\delta_{h}^{n}f(x)=\sum_{i=0}^{n}(-1)^{n-i}\binom{n}{i}f(x+ih)

of the function $f$ , defined in the interval $(a,b)$ The difference $\partial_{h}^{h}f(x)$ is, up to a factor independent of the function, a dd of order $n$ ,

\hat{o}_{h}^{n}f(x)=n!h^{n}[x,x+h,x+2h,\ldots,x+nh;f].

Definition. - We will say that the function $f$ , defined in the interval $(a,b)$ is convex, non-concave, polynomial, non-convex resp. concave of order $n(\mathrm{~J})$ , or in Jensen's sense, in ( $a,b$ ) following that inequality
$\hat{o}_{h}^{n+1}f(x)>,\geqq$ , =, sesp. $<0,\quad x,x+(n+1)h\in(a,b),\quad h>0$ is satisfied.

All these functions are functions of order n (J).
It follows immediately that, for these functions, inequality (1) is satisfied, provided that the points $x_{1},x_{2},\ldots,x_{n+2}$ are divided rationally, that is to say that the ratios $\frac{x_{n+2}-x_{i}}{x_{i}-x_{1}}$ be rational.

The definition can be extended to any set E such that with $x_{1},x_{2}\in\mathrm{E}$ we always $\frac{x_{1}+x_{2}}{2}\in\mathrm{E}$ . Thus we can extend the definition to any set of rational points, but in this case, according to the previous remark, these functions are of order n in the ordinary sense (of chap. I).

Any convex, non-concave, etc. function of order $n(J)$ on $E$ is convex, non-concave, etc. of order $n$ in the ordinary sense on any subset of $E$ whose points divide rationally.
24.—Any convex, non-concave, … etc. function of order $n$ in the ordinary sense $(a,b)$ is obviously convex, non-concave, etc., of order $n(J)$ In $(a,b)$ . If $n=0$ The converse is naturally true since the two definitions coincide, but if $n>0$ The converse is only true if certain restrictive assumptions are made about the function. $f$ This is the case if the function is continuous in $(a,b)$ or, more generally, if the function is bounded in $(a,b)$ JLWV Jensen [27] demonstrated that, for the non-concave function of order $1$ , it suffices even that it be bounded above. P. Tortorici [60 a] gave a new proof of this property. The converse property is also true when the function is measurable in $(a,b)$ , as demonstrated for $n=1$ H. Blumberg [10] and W. Sierpiński [54b]. W. Sierpiński [54c] even demonstrated that, for $n=1$ and the non-concave function, it suffices that there exists a measurable function $\varphi$ such as $f=\varphi$ . A. Ostrowski [42] demonstrates a similar property.

The study of linear functions (d) amounts to a summation at Téfade of Caveny's equality.

f(x+y)=f(x)+f(y),

(24)

whose continuous solution is $\Delta x$ , A élant, a constant. Indeed, if $f$ is linear (J) the function $f(x)$ - $f(0)$ verifies equation (24). Relative to this equality, G. Dandoux [14] demonstrates that the
solution is still Aæ provided that / is bounded above or below in a sin i interval. According to H. Lebesaus [35], S. Banach [3] and W. Sterpinski [54a], this is still the case if we assume / is measurable. M. Kac [29] ( ¹ ) recently gave an elegant proof. M. Kormes [33] showed that it is even sufficient that $f$ either bounded on a measurable set of measurement $>0$ Moreover, it can be demonstrated that, for all the stated hypotheses, it suffices that they be realized within such a small interval. Similarly, Mr. Kormes' hypothesis is applicable to all functions of order $n$ any.

Let's make a brief digression on equations of the form

\delta_{h}^{n}f(x)=0.

(25)

P. Montel [39 a] demonstrated that if $f$ is continuous and (25) is verified for all $x$ and for two fixed values $\omega_{1},\omega_{2}(\angle 0)$ of $h$ whose ratio is irrational, the function $f$ is a polynomial of degree $n-1$ This is the case even if $f$ is continuous only at n points $[47$ e]. We have studied the generalizations of equation (25) in another work [47 s].
25. - Let us say a few words about discontinuous functions of order $n(J)$ From the remarks made above, it follows that if $f$ is. of order $n(\geq 1)(J)$ , it is uniformly continuous on the set of points that rationally divide the interval $(a,b)$ and which belong to a subinterval $c\cdot(a,b)$ Thus, we find, for $n=1$ , a property of F. Berinstein [5].

Either $f$ non-concave of order $1(J)$ In $(a,b)$ closed. F. Bennstenn and G. Doetser [6] have shown that, if f is not bounded below in the interval ( $a,b$ ), it is not bounded below at any point of ( $a,b$ ). These authors also demonstrated that, if $m(x)$ is the lower bound function of $f$ and if $f\neq m$ at one point $x\in\cdot(a,b)$ , the function $f$ is not bounded at any point above, and the representative points $(x,f(x))$ curve $y=f(x)$ are dense everywhere above $y=m(x)$ The function $m(x)$ is, moreover, non-concave of order 1 if / is bounded below.

⁰⁰footnotetext: (1) Les démonstrations de MM. W. Srerpinski et M. Kac sont particulièrement importantes puisqu’elles n’emploient pas l’axiome de Zermelo. Au contraire, les résultats qui suivent ici sont démontrés à l’aide de cet axiome (plus exactement à l’aide du fait qu’un ensemble peut être bien ordonnó).

Otherwise, the representative points ( $x,f(x)$ ) are everywhere dense in the band between the ordinates a and b. G. Hamel. [20] also notes that, if $f$ is linear (J) in ( $-\infty$ , + $\infty$ ), the representative points are everywhere dense in the plane.
G. Hamel [20] constructed the most general linear function (J) by solving equation (24). Up to an additive constant, this function is of the form

\left.\int r_{1}\alpha+r_{2}\beta+\cdots\right)=r_{1}f(\alpha)+r_{2}f(\beta)+\cdots,

Or $r_{1},r_{2},\ldots$ are rational numbers, $\alpha,\beta,\ldots$ the elements of a base H0 of real numbers, therefore numbers such that all $x$ can be represented in a unique way in the form

x=r_{1}^{\alpha}+r_{2}^{\beta}+\cdots,\quad\alpha,\beta,\cdots\in\mathcal{H},

with a finite number of rational numbers $r_{1},r_{2},\ldots$ no, null and finally $f(\alpha),f(\beta),\ldots$ are arbitrary given numbers.
MH Ingraham [25] generalized this result, giving the general solution of equation (25).
26. - By considering only dd on points equidistant from the function / defined in an interval ( $a,b$ ), which we can assume to be open, we can still define an nth bound $\Delta_{n}^{*}$ , which is, by definition, the maximum of the absolute value of dd $\frac{1}{n!h^{n}}\delta_{n}^{n}/(x)$ in the meantime $(a,b)$ . If $A_{n}^{*}$ is, finished, the function $f+A_{n}^{2}x^{2}$ is non-concave of order $n-1(J)$ .

Any order dd $n$ of $t$ , taken at points that are rationally divisible, is contained between — $\Delta_{n}^{*}$ And $\Delta_{n}^{*}$ , from which we deduce that if $f$ is continuous $A_{n}^{*}=A_{n},A_{n}$ being, the nth boundary defined at $n^{\circ}9.11$ It also follows that, if $n\geq 2$ and if $f$ is bomed or measurable and $\Delta_{n}^{*}$ is finite, the function is continuous,
S. Saks [51 b] notes that, if the function / is differentiable, and

\lim_{h\rightarrow 0}h^{-3}2_{2h}f(x-3h)\geq 0,\quad x\in\cdot(a,b),

it is non-concave of order 2 in $(a,b)$ . S. Verblunsky 63 a] demonstrates, moreover, that, if / is differentiable in the interval formed ( $a,b$ ), and if the generalized third derivative

\lim_{h\rightarrow 0}(2h)^{-3}r_{2h}^{3}f(x-3h)=q(x),\quad\text{ ae }(a,b)

exists, we have, for $a<c<d<b,3h_{1}=d-c$ ,

\min_{(c,d)}p(x)\leqq\left[\left(2h_{1}\right)^{-3}\delta_{2}^{3}h_{1}f\left(x-3h_{1}\right)\right]_{x=c}\leq\max_{(c,d)}p(x),

and finds a similar property for the fourth-order difference $\delta\frac{1}{h}f(x-2h)[63\mathrm{~b}]$ .

Note also that, according to A. Marchauo [36], if f is bounded in the closed interval $(a,b)$ And $\delta_{h}^{n}f(x)\rightarrow 0$ uniformly with respect to $x$ The function is continuous. Likewise, if $h^{-n}\hat{\delta}^{n}/f(x)\rightarrow g(x)$ uniformly, $f$ has a continuous nth derivative equal to $g(x)$ These results remain valid if we assume the function $f$ measurable [47 q], as demonstrated, for the second property, also by H. Wittmey [64].
27. - S. Bernstern [7 b] considers the functions $f$ which verify the inequalities $\delta_{h}^{n}f(x)\geq 0,h>0,n=1,2,\ldots$ in an interval (a, b). These are completely monotonic functions in the sense of No. 17 ( ¹ ). These functions are therefore constant and indefinitely differentiable. Moreover, S. Berenstern demonstrates that they are analytic and developable in powers of $x$ - a in an interval $(a-p,a+p),p\geq b-a$ . For a function to be the difference between two completely monotonic functions in ( $a,b$ ), it is necessary and sufficient that it be analytical and developable according to the powers of $x-a$ In $(a-p,a+p)$ , with $p\geq b-a$ The functions that are beautiful that $\delta_{h}^{n}f(x)$ is of invariable sign for each positive integer $n$ , are ineove analybic and developable according to the powers of $x-a$ In $(a-p,a+p)$ , with $\rho\geq\frac{b-a}{4}$ S.
Bennstern [7e] made a systematic study of monotonic complementary functions in $(-\infty,0)$ and determined those which take, with their first n or with all their derivatives, given values for $x=0$ This problem is closely related to the moment problem since a completely monotonic function in ( $\cdots\infty$ , 0) is representable by an integral of the form

f(x)=\int_{0}^{+\infty}e^{tx}d\varphi(t)

oì φ is non-decreasing.

⁰⁰footnotetext: (1) M. 3. Bernstenn les appolle absolument monotones.

28. - We can also generalize functions of order n in another way. Consider a finite and closed interval $(\alpha,\beta)$ containing the bounded set E. Let

f_{0},f_{1},\ldots,f_{n},\ldots

(26)

a finite or infinite sequence of functions defined in $(a,b)$ These functions are said to form a basis when a linear combination $c_{0}f_{0}+c_{1}f_{1}+\ldots+c_{n}f_{n}$ with constant coefficients $c_{i}$ , is completely determined by its values in $n+1$ distinct points, and this regardless of $n$ and the $n+1$ points considered.

Let us designate by $\mathrm{D}\left(x_{1},x_{2},\ldots,x_{n+1}\right)$ the determinant

\left|f_{0}\left(x_{i}\right)f_{1}\left(x_{i}\right)\cdots f_{n}\left(x_{i}\right)\right|,i=1,2,\ldots,n+1

and by $\mathrm{D}\left(x_{1},x_{2},\ldots,x_{n+2};f\right)$ the determinant

\left|f_{0}\left(x_{i}\right)f_{1}\left(x_{i}\right)\cdots f_{n}\left(x_{i}\right)f\left(x_{i}\right)\right|,i=1,2,\ldots,n+2,

$f$ being any function. For the functions (26) to form a basis, it is necessary and sufficient that we have $\mathrm{D}\left(x_{1},x_{2},\ldots,x_{n+1}\right)=0$ For $x_{1},x_{2},\ldots,x_{n+1}\in(\alpha,\beta)$ and distinct and for $n=0,1,\ldots$ We will also say that the functions (26) form a system (T) if

\begin{gathered}\mathrm{D}\left(x_{1},x_{2},\ldots,x_{n+1}\right)>0,x_{1}<x_{2}<\cdots<x_{n+1},\\ x_{1},x_{2},\ldots,x_{n+1}\in(\alpha,\beta)n=0,1,2,\ldots\end{gathered}

If the functions (26) are continuous and form a basis, these functions form a system (T), provided that the sign of some of these functions is possibly changed.

In what follows we only consider sequences (26) forming a system (T).

Definition. - We will say that the function $f$ is convex, nonconcave, polynomial, nonconvex, concave (T), or with respect to the functions $f_{0},f_{1},\ldots,f_{n}$ , on E , if the inequality

\mathrm{D}\left(x_{1},x_{2},\ldots,x_{n+2};f\right)>,\geqq,=,\leqq,<0

is verified, regardless of $x_{1}<x_{2}<\ldots<x_{n+2},x_{1},x_{2},\ldots,x_{n+2}\in\mathrm{E}$ [47 f].

To simplify the language we will say that all these functions are (T) functions.

Convexity (or concavity) (T) essentially expresses the property that $f_{0},f_{1},\ldots,f_{n},f(ou-1)$ form a system (T). Conversely, in a system (T) every function is convex (T) with respect to the functions that precede it in the sequence (26).

A polynomial function (T) reduces to the values on E of a linear combination $c_{0}f_{0}+c_{1}f_{1}+\ldots+c_{n}f_{n}$ functions (26). Moreover, we can geometrically characterize a function (T) using these linear combinations, just as in the particular case $f_{0}=1,f_{1}=x,\ldots,f_{n}=x^{\prime\prime},\ldots$ which we have hitherto eluded.

For any function (T) to be bounded on all $\mathrm{E}_{2}\subset\cdot\mathrm{E}_{1}$ If necessary and sufficient, the functions $f_{10},f_{1,\ldots},f_{n}$ be it, bomess on your $E_{1}\subset$ E Si $a,b$ , EE , the property extends to the set E. In particular, for the functions of a system (T) to be bounded, it is necessary and sufficient that the first function fo be bounded [47f]. We have an analogous property with regard to the continuity of a function (T) on a $E_{1,}\cdots\cdot E_{1}$ if $n\geq 1$ In particular, the functions of the system (T) are continuous in the interval ouport ( $*,\beta$ ) if, and only if, the first two functions $f_{0},f_{1}$ soni continues in this interval. We have the proper way, by taking, instead of the property of continuity, the property of being at first dd bomée el mème, sous corbaines condilions restrictives, d'être à kème (k¿1) dd bomée [47 f].

The preceding results can be extended, under certain conditions, to the somewhat more general case where $f_{0}$ cancels out a finite number of times in $(\alpha,\beta)$ If, for example, $n\geq 1$ And / ${}_{0},f_{1}$ are continuous, we can first consider the intervals where $f_{0}$ does not cancel out, and then the intervals that contain a zero of $f_{0}$ and where $f_{1}n$ e does not cancel out, intervals that exist, since $f_{0},f_{1}$ cannot cancel each other out at the same time. In these intervals, it suffices to reverse the roles of the functions. $f_{0},f_{1}$ .

The functions ( $T$ ), in the case $n=1$ , onli été informed by E. Phragmen and E. Lindelöf [45] who considered the case

f_{0}=\cos x,\quad f_{1}=\sin x

which is important in the theory of functions of a complex variable. The functions (T) in the interval $(\varepsilon,\pi-\varepsilon),\varepsilon>0$ are, then, those for which the determinant,

\left|\begin{array}[]{ccc}\cos x_{1}&\sin x_{1}&f\left(x_{1}\right)\\ \cos x_{2}&\sin x_{2}&f\left(x_{2}\right)\\ \cos x_{3}&\sin x_{3}&f\left(x_{3}\right)\end{array}\right|,\quad x_{1}<x_{2}<x_{3},\quad x_{1},x_{2};x_{3}\in(\varepsilon,\pi\ldots\varepsilon)

does not change sign. From what we have said, it follows that
$f$ is continuous and even satisfies an ordinary Lieschitz condition in every completely interior interval.
A. Winternitz [65] generalized these results, considering the case where $f_{0},f_{1}$ are bounded at first variation within an interval $(\alpha,\beta)$ and demonstrated that all functions $(T)$ In $(\alpha,\beta)$ is bounded at first variation in every interval $c\cdot(\alpha,\beta)$ and therefore admits a left-hand derivative and a right-hand derivative at a point $x\subset(x,3)$ which are of bounded variation (of order 0) in high interval $c\cdot(\alpha,\beta)$ .

The case of first-order functions in $(0,1)$ comes back to the case $f_{0}=1-xf_{1}=x$ J. Radon [49] has already demonstrated these results in the case $f_{0}=(1-x)^{p},f_{1}=x^{p},p\geqslant 1$ In $(0,1)$ The results were also found by G. Polya [46 b] in case (27) and by G. Valirov [61 b] in the general case, in a form which returns to that of A. Winternitz [65]. J. Radon [49], in the case which he elucidated, and A. Winternita [06] in the general case, extended the results of W. Blaschke and G. Prok [9] (see No. 11) on the representation of these functions by a Stiejljes integrate.

We can note that the Lonobiois of s. Rabon [40] still enjoy the property that one can divide Pinkervalle (0,1) into two intervals in which the function is monotonic. Moreover, the power ( $p=1$ ) of a non-convex, first-order, non-negative union in $(0,1)$ is non-convex in the sense of J. Radon. It is the same, likewise, of the power of a non-concave function of order 1 and non-positive in (0,1) if $p$ is an odd positive integer.

Let us suppose, more generally, that $f_{0},f_{1},\ldots,f_{n}$ are linearly independent solutions of a linear differential equation of order $n+1$ ,

L(y)\equiv y^{(n+1)}+\varphi_{1}y^{(n)}+\varphi_{2}y^{(n-1)}+\cdots+\varphi_{n}+1y=0,

where the coefficients of continuous functions in $(\alpha,\beta)$ . Aloxs houte function (T) is at nth dd bounded on ton $\mathrm{E}_{1}\in$ F.

The case of Chapter I corresponds to the equation $y^{(n-1)}=0$ Case (27) in equation $y^{\prime\prime}+y=0$ The case of J. Radon with the equation $x(1-x)y^{\prime\prime}+(p-1)(2x-1)y^{\prime}-p(p-1)y=0$ within an interval $c\cdot(0,1)$ .

If the function (T) has in this case a ( $n+1$ ) 4. In derivative, noncavity (T) is expressed by inequality $L(j)\geq 0$ The condition $L(j)>0$ ost susinante for the convexile (T).

Let us also mention that S. Kakeya [30 a] determined the necessary and sufficient conditions for the existence of a function f $(n+1)$ times differentiable, such that $\mathrm{L}(j)\geq 0$ and engaging, with its $m$ first derivatives, the given values

f^{(i)}(\alpha)=a_{i},\quad f^{(i)}(\beta)=b_{i}\quad i=0,1,\ldots,n_{s}

This generalizes a prolongation problem examined in no. 6.
29. - To conclude with functions of one variable, let us say that E.F. Beckenbach [4] gave an even more extensive generalization. This generalization is as follows:

Let us consider, in an open interval ( $a,b$ ), a family of functions $\mathrm{F}=\mathrm{F}(x;\lambda,\mu)$ depending on two parameters $\lambda,\mu$ and meeting the following conditions: $1^{\circ}$ each function of the family is continuous in $(a,b);20$ there is one and only one function of the family that satisfies the system

F\left(x_{1};\lambda,\mu\right)=y_{1},\quad F\left(x_{2};\lambda,\mu\right)=y_{2}

(28)

for any pair of distinct points $x_{1},x_{2}$ , of $(a,b)$ and for everything $y_{1},y_{2}$ .
EF Beckenbach then said that the function $t$ , defined in the open interval ( $a,b$ ), is a sub- function $F^{\prime}$ yes, whatever the points $a<x_{1}<x_{2}<b$ we have $f(x)\leqq\mathrm{F}_{12}(x)$ the second member designating the function $\mathrm{F}(x;\lambda,\mu)$ which verifies the equalities (28) for $y_{1}=f\left(x_{1}\right),y_{2}=f\left(x_{2}\right)$ The main result obtained is that any function subF is continuous in $(a,b)$ .

We can generalize further by considering convexities with respect to a family $F\left(x;\lambda_{1},\lambda_{2},..,\lambda_{n+1}\right)$ depending on $n+1$ parameters $\lambda_{1},\lambda_{2},\ldots,\lambda_{n+1}$ such as the system

\mathrm{F}\left(x_{i};\lambda_{1},\lambda_{2},\cdots,\lambda_{n+1}\right)=y_{1}\quad i=1,2,\cdots,n+1

always have a single function $F$ as a solution. We can also consider functions defined only on an arbitrary set E belonging to ( $a,b$ ).

CHAPTER IV

CONVEX HONCTIONS
OF TWO OR MORE VARIABLOS

30.
- •
  
  A pseudo-polynomial of order ( $m,n$ ) is not necessarily boné el eneore less continu in R. But, if it is boné resp. continu on a lattice of order ( $m+1$ , $n+1$ ), it is a bounded function, respectively continuous in R. Similarly, if such a pseudopolynomial admits partial derivatives $f_{x^{r}}^{(r)},f_{y^{s}}^{(s)}$ resp. of continuous partial derivatives $f_{x^{r}}^{(r)},f_{y^{s}}^{(s)}$ on an order network ( $\left.m+1,n+1\right)$ It has partial derivatives in R. $f_{x^{1/2}y^{\beta}}^{(\alpha+\beta)}$ resp. of continuous partial derivatives $1^{\left(\alpha+\beta^{\beta}\right)},\operatorname{avec}x\leqq r,\beta\leqq s$ .

In the study of functions $f$ In $R$ it is convenient to introduce the number

\mathrm{A}_{m,n}=\underset{(\mathrm{R})}{\max}\left|\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};\right|

which we call the order boundary $(m,n)$ of the function / in R. If $\Delta_{m,n}$ is finished the function is at dd of order ( $m$ ; n) bounded.

In particular, a function with dd of order ( $m,n$ ) bounded is at mane dd bounded with respect to $x$ for all values of $y$ , but the converse is obviously not true. A boone function is a function ìdd of order $(0,0)$ bomée.

A pseudo-polynomial of order ( $m-1,n-1$ ) is obviously in dd order $(m,n)$ Bomée, but its order dd $\left(m_{1},n_{1}\right)<(m,n)$ may not be bounded in R. So that it may be so for all order $\leq(m,n)$ , it is necessary and sufficient that its dd of order (m, 0) and of order $(0,n)$ are, bounded on a network of order $(m,n)$ Furthermore, if the pseudo-polynomial is dd of order $\left(m_{1},n_{3}\right)$ el d'oráre $\left(m_{2},n_{1}\right)$ limited, with $m_{1}\geq m_{2},n_{1}\geq n_{2}$ , it is, to give ( $m^{\prime},n^{\prime}$ ) bounded, with $m_{1}\geq m^{\prime}\geq m_{3},n_{1}\geq n^{\prime}\geq n_{2}$

Any function with dd of order ( $m,n$ ) bounded is the sum of a function ayont all its dd of order $\leq(m,n)$ bounded eb of a pseudo-polynomial of order ( $m-1,n-1$ Done, so that all the order dd $\leqq(m,n)$ of a function are bounded, it is necessary and sufficient that its dd of order ( $m,n$ ) is bounded and its dd of order $(m,0)$ and order $(0,n)$ are bounded on a network of order ( $m,n$ ).

A function that is continuous with respect to one of the variables and also continuous with respect to the other is continuous in R. In particular, a function of order d is continuous. $(1,0)$ bounded and continuous with respect to y is continuous in R. It follows that any function with dd of order ( $m,n$ ) bounded, with $m\geq 1,n\geq 1$ , continues on a network of order ( $m,n$ ), is continuous in R.

If a function $f$ to dd order $(m,n)$ bounded to a partial derivative $f_{x,}^{\prime}$ this derivative is a function with dd of order $(m-1,n)$ bounded. We deduce that, if in addition the derivatives $f_{a^{m-1}}^{(m-1)}$ , $f_{y^{m-1}}^{(n-1)}$ are continuous on a network of order ( $m,n$ ), the function has partial derivatives $f_{ay^{3}}^{(r+s)},r\leq m-1,s\leq n-1$ , to dd in order ( $m-r,n-s$ ) bounded in R.
P. Montex [39 a] and A. Marchaud [36] also demonstrated that, for a function with dd of order $(m,0)$ and order $(0,n)$ bounded, all partial derivatives $\int_{r^{\prime}y^{s}}^{(r+s)}$ exist and are continuous in R., provided that $\frac{r}{m}+\frac{s}{n}<1$ It follows that, if a function is at dd of order ( $m$ - m', n) and of order ( $m,n$ - $n^{\prime}$ ) bounded, it is also at dd of order ( $m-r,n-s$ ) limited, provided that.

\frac{r}{m^{\prime}}+\frac{s}{n^{\prime}}>1,\quad m\geq m^{\prime}>0,\quad n\geq n^{\prime}>0,\quad r\leq m^{\prime},\quad s\leq n^{\prime}.

$1. - Let's now move on to the definition of convexity.
First of all, the following definition is necessary.
Definition. - The function $f$ is said to be convex, non-concave, polynomial, non-convex, concave of order ( $m,n$ ) in rectangle R if the inequality

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+2};f\\ y_{1},y_{2},\ldots,y_{n+2}\end{array}\right]>,\rightleftarrows,=,\leq,<0

is satisfied, regardless of the points $\left(x_{i},y_{j}\right),i=1,2,\ldots,m+2$ ,
$i=1,2,\ldots,n+2$ , belonging to R and forming the nodes of a network of order ( $m,n$ ).

All these functions are order functions ( $m,n$ ).
The -1 values of $m$ And $n$ are not excluded. A convex, non-convex, etc. function of order $(m,-1)$ is a function that is convex, non-concave, ..., etc., of order $m$ compared to $x$ shames the values of $y$ .

A polynomial function of order ( $m,n$ ) is a pseudo-polynomial of order ( $m,n$ ). If $f$ is of order ( $m,n$ ) and if

\left\lfloor\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+2}\\ y_{1},y_{2},\ldots,y_{n-2}\end{array};t=0,\right.

it reduces to a pseudo-polynomial of order ( $m,n$ ) in the rectangle

\left(x_{1}\leqq x\leqq x_{m+2};y_{1}\leqq y\leqq y_{n+2}\right).

A geometric definition, analogous to that given in the case of a single variable, can be obtained using pseudopolynomials. Consider the pseudopolynomial

\mathrm{P}\left(\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+1}\\ y_{1},y_{2},\ldots,y_{n+1}\end{array};f\binom{x}{y}\right.

Non-concavity or convexity of order ( $m,n$ ) is expressed by the fact that the function must be, at every point of the rectangle

\left(x_{i}<x<x_{i+1},y_{j}<y<y_{j+1}\right)

not below or above, or not above or below, the following pseudo-polynomial $m+n-i-j$ is even or odd. This also applies to $i=0,m+1,j=0,n+1$ , in the rectangles

\begin{array}[]{ll}\left(a\leqq x<x_{1},\right.&\left.c\leqq y<y_{1}\right),\\ \left(a\leqq x<x_{1},\right.&\left(x_{n+1}<x\leqq b,\quad c\leqq y<y_{1}\right)\\ \left(x_{n+1}<x\leqq b,\quad y_{n+1}<y\leqq b\right).\end{array}

It follows immediately that if a function of order $(m,n)$ is bounded on a network

x=x_{i},\quad y=y_{n},\quad i=1,2,\ldots,m+2,\quad j=1,2,\ldots,n+2,

order ( $m+2,n+2$ ), it is bounded within the rectangles

\left(a\leq x\leq b,\quad y_{1}\leq y\leq y_{n+2}\right),\quad\left(x_{1}\leq x\leq x_{m+2},\quad c\leq y\leq d\right)

It can still be easily seen that, when the order dd ( $m+1,n$ ) And ( $m,n+1$ ), of an order function ( $m,n$ ), are bounded on a network

x=x_{i},\quad y=y_{j},\quad i=1,2,\ldots,2m+2,\quad j=1,2,\ldots,2n+2,

diorder ( $2m+2,2n+2$ ), these dd are bounded in the reeLauge
(30)

\left(x_{m+1}\leq x\leq x_{m+2},y_{n+1}\leq y\leq y_{n+3}\right).

So that, moreover, the dd of everything, order! ( $m+1,n+1$ ) are, boxed in the rectangle (30), it is necessary and sufficient that the dd of order ( $m+1,0$ ) and order ( $0,n+1$ ) are bounded on a network of order ( $m+1,n+1$ ) whose nodes belong to (30).

The differentiability properties of order functions ( $n,n$ ) result from those of the fonebions is dd homées. If the derivation (rens), $r\leq m+1,s\leq n+1,r+s<m+n+2$ exists, it is a union of order ( $m-r,n-s$ ), presupposing the same convexity. When $f_{a^{n+1}}^{(n+n+2)}$ It exists; it is non-negative. The condition $f_{(m+n+2)}^{(m+1)}\geq 0$ is, moreover, necessary and sufficient for non-conviviality, and $f_{25+n+2)}^{(m+1)}\geq 0$ is sufficient for the convexity of order ( $m,n$ ) in R.

The (m, n) order functions can also be defined in any domain D.

We can also define functions of order (ni, n) by simply imposing the condition that, for every group of $n+2$ value $y_{1},y_{2},\ldots,y_{n+2}$ of $y$ the function of $x,\left[y_{n},y_{2},\ldots,y_{n+2};f\right]$ either of order $n$ ot, for any group of $m+2$ values $x_{1},x_{2},\ldots,x_{m+2}$ of $x$ , the suspension of $y,\left[x_{1},x_{21}\ldots,x_{m+2};f\right]$ either of order $m$ We will not dwell on this generalization; we will only consider the partition cases. $m=0,n=0$ to the following numbers.
32. - Let's make the change of variables

x=\alpha x^{\prime}+\beta y^{\prime},\quad y=\gamma x^{\prime}+\delta y^{\prime},\quad\alpha\hat{b}-\beta y,0.

(31)

We will say that this transformation defines a direction. This means that we have adopted a new system of axes Oæ'y'. In particular, the initial system Oæy defines the direction. We can assume, without restricting the generality, that $\alpha^{2}+\gamma^{2}==\beta^{2}+\mathrm{o}^{2}=1$ .

The change of variables (31) transforms the function / into a function $f_{1}\left(x^{\prime},y^{\prime}\right)$ of $x^{\prime}$ And $y^{\prime}$ The dd of $f$ in the direction $t$ are then the $d.d$ . of $f_{1}$ as a function of $x^{\prime};y^{\prime}$ We can then
define functions that exhibit a property of convexity in any direction, but these functions revert, through a simple change of variables, to the functions previously defined.

Or we can also consider functions that exhibit convoxity in two or more directions, whether distinct or not. For example, a function that is polynomial (not necessarily of the same order) in two directions generally reduces to a polynomial in $x$ And $y$ This is surely the case if the directions are $t_{0}$ And $t$ with $\alpha\beta\gamma\delta\neq 0$ , up to a further change of variables. If, in addition, the function is polynomial of order ( $m,n$ ) in the direction $t_{0}$ and polynomial of order ( $m^{\prime},n^{\prime}$ ) in the direction $t$ it reduces to a polynomial of degree $\leqq m+n+m^{\prime}+$ - $n^{\prime}+2$ The only polynomial functions of order $(m,n)$ in all directions are polynomials of degree $<m+n+1$ but we can find polynomials of degree $m+n+1$ polynomials of order ( $m,n$ ) in an infinite number of directions [47 s].

Let us leave aside these generalities and return to the simplest cases, which are, moreover, the most important.
P. Mondex [39 e] studied the non-concave functions of order 1 with respect to a for all values of $y$ and non-concave of order 1 with respect to $y$ for any value of $x$ , calling them doubly convex, functions which have also been considered by N. Krutkos [34].

More generally, we will say that a function $f$ is of order $n$ in $x$ responsible for order $n$ in $y$ if it is of order $n$ with respect to x for any value of $y$ resp. d'ordeo $n$ compared to $y$ for any value of $x$ We will also say that a function is doubly of order $[m,n]$ if it is of order $m$ in $x$ and order $n$ on $y$ Note that in these definitions, the function is not required to exhibit the same convexity characteristic of function m with respect to $x$ for all values of y or that it exhibits the same convexity characteristic of order n with respect to y for all values of $x$ In particular, if / is non-oonoavo of order n with respect to $x$ (ear relation to $y$ ) for any value of $y[$ of $x]$ no, let's return to a function of order ( $n,0$ ) [of order ( $0,n$ )]. It is clear what is meant by a doubly nonconcave function of order [m, n]. P. Montel's functions are doubly nonconcave of order [1,1].

It is shown that any function of order $m$ in $x$ el continues with respect to y is at dd order ( $m$ , 0) bounded in a rectangular end c . R. H is sufficient even, for it to be so, that $f$ either bounded in $y$ in any interval $c\cdot(c,d)$ , for any value $x\in\cdot(a,b)$ A doubly ordered function $[m,n],m\geq 1,n\geq 1$ is therefore in order dd ( $m,0$ ) and order ( $0,n$ ) bounded, therefore is continuous and, according to P. Montel [39 a], has a partial derivative $f_{a^{r}y^{s}}^{(x+s)}$ eontinue, pourru que $\frac{F}{m}+\frac{s}{n}<1$ , in any completely interior rectangle.

The double non-coneaves functions of order $[0,0]$ were examined by WH Young and GC Young [66]. If ( $x_{n},y_{n}$ ) is, a point of R , these functions enjoy the property that, if a sequence of points $\left(x_{p},y_{p}\right),p=1,2,\ldots$ from Retond towards $\left(x_{1},y_{0}\right)$ in such a way that one constantly $x_{p}>x_{0},y_{p}>y_{0}$ Or $x_{p}<x_{0}$ , $y_{p}<y_{0},f\left(x_{p},y_{p}\right)$ end towards a limit $\geq f\left(x_{0},y_{0}\right)$ Or $\leqq f\left(x_{0},y_{0}\right)$ These authors also studied the case where, in addition, the function is non-conoave of order (0,0).

We can also consider doubly polynomial functions of order $[m,n]$ which are polynomials of order $m$ in $x$ and polynomials of order n in y. These are obviously polynomials of degree $m$ in $x$ degree of $n$ on $y$ , therefore what can be called a polynomial of degree (m, n).
33. - We can finally define functions in a domain D, which exhibit certain convexity properties in all directions.

In particular, we will say that / is completely of order n in D if it is of order $n$ on any straight line intersecting the domain D, in other words it is of order ( $n,0$ ) in any direction. When n is even, there is no need to distinguish between convexity and concaveness. On the contrary, if n is odd, we can define completely convex, non-concave, ... functions of order n.

A completely orderly function $n$ is, in particular, a doubling of order $[n,n]$ , therefore it is continuous at every point inside D if $n>0$ A function of order 0 is bounded in all $D_{1}c\cdot D$ A function completely of order 'n' has partial derivatives of order . $<n$ continuous at every
interior point. If, in particular, the partial derivatives of order $n+1$ exist, the function

\sum_{i=0}^{n+1}\binom{n+1}{i=0}\cos^{n-i+1}\alpha\sin^{i}\alpha\frac{\partial^{n+1}f}{\partial x^{n-i+1}\partial y^{i}}

(32)

must have an invariable sign on any straight line making the angle $\alpha$ with the axis $0x$ If, in addition, the function is completely non-concave of order $n$ odd, for all $x,y$ The polynomial (32) must be non-negative. In particular, the conditions

f_{x^{2}}^{\prime\prime\prime}\geq 0,\quad f_{x^{2}}^{\prime\prime}f_{y^{2}}^{\prime\prime}-\left(f_{xy}^{\prime\prime}\right)^{2}\geq 0

are necessary and sufficient for complete non-concaveness, and

f_{x^{2}}^{\prime\prime\prime}>0,\quad f_{2}^{\prime\prime}f_{y^{2}}^{\prime\prime}-\left(f_{xy}^{\prime\prime}\right)^{2}>0

are sufficient for the complete first-order convexity of $f$ assuming, of course, that second derivatives exist.

Jensen's inequality [27] can be extended to completely non-concave functions of order 1. We have the inequality, established by F. Sibrani [53],

f\left(\frac{\Sigma p_{i}x_{i}}{\Sigma p_{i}},\frac{\Sigma p_{i}y_{i}}{\Sigma p_{i}}\right)\leqq\frac{\Sigma p_{i}f\left(x_{i},y_{i}\right)}{\Sigma p_{i}},\quad p_{i}>0.

(33)

EJ McSuane [38] generalized this inequality for two or more variables, as in the case of a single variable.

A completely polynomial function of order $n$ reduces to a polynomial of degree $n$ in D.
34. – If $f$ , g are two non-concave functions of order ( $m,n$ ), cf and $f+g$ , Or $c$ is a positive constant, are still non-concave functions of order ( $m,n$ The limit of a convergent sequence of non-concave functions of order $(m,n)$ is still nonconcave of order ( $m,n$ Moreover, any continuous function in R exhibiting one or more well-defined convexity properties is the limit of a uniformly convergent sequence in R of polynomials exhibiting the same convexity properties, as shown by the polynomials of S. Barnsterin in soft variables.

\left.\frac{1}{(b-a)^{p}(d-c)^{p}}\sum_{i=0;0;0}^{\eta}\binom{p}{i}\binom{q}{j}f_{i}^{\prime}a+i\frac{b-a}{p},c+j^{d-c}\frac{d-c}{q}\right).

M. Nicoresco [41] demonstrated, as in the case of a variable, that a family of doubly non-concave functions of order 1 and equally bounded above in $D$ is normal inside do $D$ and deduces that any non-decreasing sequence of doubly non-concave functions of order 1 that converges at a point e. D, converges uniformly inside D.
J.L.W.V. Jensen [27] already noted that a doubly non-concave function of order 1 is not necessarily completely of order 1.
F. Riesz [50-6] generalizes non-concave functions of order 1 of one variable by subharmonic functions of two or more variables. A function $f$ is subharmonic in D if, $\mathrm{D}_{1}$ being any subdomain of D, it is, at every point of $D_{1}$ , not above any harmonic function that it does not exceed at any point on the boundary of $\mathrm{D}_{1}$ If the subharmonic function is continuous, or more generally if the following integral exists, it is characterized by the fact that it remains at every point e . D, not above its average value taken over a circumference having co as its opposite point and belonging to D. The subharmonic function $f$ is therefore characterized by inequality

f(x,y)\leq\frac{1}{2\pi}\int_{0}^{2\pi}f(x+r\cos\theta,y+r\sin\theta)d\theta

whatever $(x,y)\in\cdot\mathrm{D}$ And $r<$ (if D is closed) $\leqq$ ) that the distance from the point $(x,y)$ at the border of D.

The study of subharmonic functions is beyond the scope of this book. We will limit ourselves to recalling a few results related to convex functions.
P. Montel [39b] demonstrated that every doubly non-concave function of order 1 is subharmonic. This property is not true for doubly 1st-order functions.
P. Montel [39b] and T. Rado [48a] demonstrated that the necessary and sufficient condition for $f$ either subharmonic is that $e^{f+\alpha x+\beta y}$ that is, for all values of the constants $\alpha,\beta$ . P. Montel [39 c] showed that the property also remains for doubly non-concave functions of order 1 and completely non-concave functions of order 1.

Soil $F(t)$ a continuous function in $(-\infty,+\infty)$ And $/$ continued
in DS Saks [51 a] demonstrates that if $\mathrm{F}(f+\alpha x+\beta y+\gamma)$ is subharmonic for all values of the constants $\alpha,\beta,\gamma$ , the function $F$ is non-concave of order 1 and: $1^{\circ}$ or $F$ is a constant; $2^{\circ}$ or $f$ is harmonic; $30^{\circ}$ or F is non-decreasing and $f$ is subharmonic; $4^{\circ}$ or finally $F$ is non-increasing and - / is subharmonic. These properties remain for doubly non-concave functions of order 1 and completely non-concave functions of order 1. In the case $2^{\circ}$ , the harmonic function must then be a polynomial of the form $\mathrm{A}xy+\mathrm{B}x+\mathrm{C}y+\mathrm{D}$ , respectively a polynomial of the form $\mathrm{A}x+\mathrm{B}y+\mathrm{C}$ 35.
- As in the case of a variable, we can also define more general functions by considering only dd on lattices formed by equidistant lines. Let us consider the difference of order ( $m,n$ )

\delta_{h,k}^{m,n}f(x,y)=\sum_{i=0}^{m}\sum_{j=0}^{n}(-1)^{m+n-i-y}\binom{m}{i}\binom{n}{j}f(x+ih,y+jk),

which is, up to a factor independent of the function, a d of order ( $m,n$ ),

\varepsilon_{h,k}^{m,n}f(x,y)=m!n!h^{m}k^{n}\left[\begin{array}[]{l}x,x+h,\cdots,x+mh\\ y,y+k,\cdots,y+nk\end{array};f\right].

Using these differences, we can define convex, non-concave, etc. functions of order ( $m$ , $n$ ) ( $J$ ) in R, which coincide, moreover, with convex, non-concave functions, etc. of order ( $m,n$ ) in the ordinary sense, if the function is continuous in R. It can easily be seen that this is still the case if the function is bounded or linearly measurable in R. Indeed, in these cases, for each value of $y$ el, $k,y,y+(n+1)k\in(c,d)$ , the function ${\underset{a,k}{o}}_{o,n+1}^{\prime}f(x,y)$ of $x$ , is horned, or measurable, and of order $m$ , therefore of order m in the ordinary sense. We therefore have Pinegality

\left[\begin{array}[]{l}x_{1},x_{2},\ldots,x_{m+2}\\ y,y+k,\ldots,y+(n+1)k;f\end{array}\right]\equiv 0,

assuming the non-concave function of order $(m,n)(J)$ The general inequality (29) then results from the fact that by fixing $x_{1},x_{2},\ldots,x_{n+3}$ , the function of $y,\left[x_{1},x_{2},\ldots x_{n+2};\right]$ is bounded, or measurable, nonconcave of order r (J). It is important to note that linear measurability cannot be replaced by surface measurability.

measurement, from the equation (24) in the interval (a,b), and oloisag do manióroquo $\chi(a)=\chi(b)=0$ We have olop $\chi(r)=0$ on tomb point. a which divides Pinbervalle raljennellement ( $a,b$ ). The sonebion

		$\displaystyle f\left(x,\frac{c+a}{2}\right)=x(x),\quad x\in(a,b),$
		$\displaystyle f\left(\frac{a+b}{2},y\right)=\chi\left(\frac{ad-bc+u(b-a)}{d-c}\right),\quad y\in(c,d),$
		$\displaystyle f(x,y)=0,x-\frac{a+b}{2},y=\frac{c+d}{2}$

mi well polynomial of orderm ( $m,m$ ) (J), $m\geq 1,n\geq 1$ , but is obviously not a proudo-polynomo.

We can also delineate functions completely in order (I). Let us consider, parlioulior, the case $n=1$ Let us consider a function $\left.f(\mathrm{P})=f\left(x_{1},x_{2}\right),\ldots,x_{m}\right)$ desinine, completely non-concave of order $1(J)$ in the convex and bounded domain D. These forchions are characterized by inequality

f\left(\frac{\mathrm{P}+\mathrm{P}^{\prime}}{2}\right)\leq\frac{1}{2}\left(f(\mathrm{P})+f\left(\mathrm{P}^{\prime}\right)\right)

Or $P,P^{\prime}\in D$ , And $\frac{P+P^{\prime}}{2}$ is the midpoint of the segment $PP^{\prime}$ These functions constitute the generalization given by JLWW IExsem [27] of the functions of a single variable that he studied. These functions: are non-concave of order 1. (J) on any line intersecting the domain D. We deduce from this, with P. Tontonca [60a], that a function entirely non-concave of order 1 (3), and bounded above in D, is completely non-concave of order 1 in the ordinary sense. If D is closed, we can observe, with II. Budarsmo [10], that it is even sufficient for the function to be bounded above, on the border F of D. In general, soib P m poini inside of D. A line passes through Po or the border Fen Pa, Pa. The ratio of the segments P. ${}_{1},\mathrm{PP}_{2}$ varies in a continuous way when the right hour around $P$ and is not generally constant, therefore necessarily takes on a reasonable value. It is reasonable that one can choose the points $\mathrm{P}_{1},\mathrm{P}_{2}$ so that $f\left(\mathrm{P}^{2}\right)$ max $\left[/\left(\mathrm{P}_{1}\right),f\left(\mathrm{P}_{2}\right)\right]$ This reasoning is flawed only if Is is a hypersphere with center $P$ , therefore at most for one point, hence the property. A. Corucer [13] remarked that, if (P) is bounded above
on a segment $\mathrm{A}_{0}\mathrm{~B}_{0}$ , it is bounded on any segment parallel to $\mathrm{A}_{0}\mathrm{~B}_{0}$ , of extremities $\in\cdot\mathrm{D}$ We can conclude that, if $f$ is measurable, it is completely non-concave of order 1 in the ordinary sense.
P. Tortorici [60 b] also notes that, if D is closed, the maximum of $f(\mathrm{P})$ , assumed to be continuous, is only attained on the boundary, unless the function is not constant. If $f(P)$ reaches its minimum on the border $F$ This is the case everywhere. $\mathrm{D}_{1}\subset\mathrm{D}$ This property is analogous to the one that defines the monotonicity pattern in the case of a single variable. This author also specifies the set of points P or min f(P) is ableint.

The surface representing the function $f(\mathrm{P})$ enjoys ownership only in every respect corresponding to $\mathrm{P}\in\cdot\mathrm{D}$ There exists a supporting hyperplane, that is to say, through such a point, one can draw a hyperplane leaving the surface not above. L. Galvan1 [19] demonstrates, in the case of two variables, that, for a tangent plane to exist at a representative point corresponding to $P$ It is necessary and sufficient that the function be differentiable on two distinct lines passing through $P$ , therefore, in particular, that the partiolles derivatives $f_{x}^{\prime},f_{y}^{\prime}$ exist at this point. Moreover, the set of points P for which the tangent plane does not exist, is of first category, therefore is formed by the union of a finite number or a countably infinite number of non-dense sets in D.

Note also that, if the set where the function continues $f+\alpha_{1}x_{1}+\alpha_{2}x_{2}+\ldots+\alpha_{n}x_{m}$ The curve that reaches its minimum is convex, regardless of the constants. $\alpha_{1},\alpha_{2},\ldots,x_{n}$ the function $f$ is completely non-concave of order 1. For the maximum, S. Saks' theorem [51 a] does not extend, since it is known that / + $x_{1}x_{1}+$ + $\alpha_{3}x_{2}+\ldots+\alpha_{m}x_{m}$ reached its maximum on the frontion of $D_{1,}$ whatever they may be, $\alpha_{1},\alpha_{2},\ldots,\alpha_{n}$ and the domain $D_{1}\subset D_{2}$ , characterizes subharmonic functions, which are not, in general, completely non-concave of order 1 [5% a].
36. - L. Galvani [19] also defined a convexity (of order 1) on an arbitrary planar set E.
ba function $f$ is non-concave on E if, for all points $\Lambda\left(x_{1},y_{1}\right),\mathrm{B}\left(x_{2},y_{2}\right),\mathrm{C}\left(x_{3},y_{3}\right)$ of E, not colinóaros, the value of $f$ in every respect ( $x,y$ ) belonging to the triangle formed ABG, is not above the plane determined by the three points representing
it $\left(x_{i},y_{i},f\left(x_{i},y_{i}\right)\right),t=1,2,3$ . Convexity, polynomiality, etc., are defined in the same way on E.

If $f$ is non-eonare with this definition it is non-concave of order 1 on every subset of E belonging to a line. L. GALFANI [19] has demonstrated a certain number of properties of cos functions. If ( $x,y),\left(x^{\prime},y^{\prime}\right)$ are two points of F, belonging to the triangle $\triangle BC$ , the point $(x,y,f(x,y))$ is at the axterior of the three-point angle formed by the point ( $\left.x^{\prime},y^{\prime},\left(x^{\prime},y^{\prime}\right)\right)$ and the points A, B, C. The function is defined in any triangle ABC, provided that the points where the values of / are taken relative to E.

If E reduces to a domain D, these functions are identical to completely non-convex functions of order 1.
37. - To conclude, we will also point out some generalizations of convex functions of several variables.
G. Aumann $[2\mathrm{a}]$ defines the non-coneave sonelion $f\left(x_{1},x_{2},\ldots,x_{m}\right)$ in the rectangle $a\leqq x_{i}=b,i=1,2,\ldots$ , but with the help of the means he introduced [2 b]. An average $M\left(x_{1},x_{2},\ldots,x_{n}\right)$ is defined by the following properties:

10 M is continuous and symmetrical in $x_{2}$ belonging to J interval $(a,b)$ ;

20 there is a $p>0$ such as if $x_{1}\geqslant x_{2}\ldots\ldots x_{n}$ we have

\begin{gathered}x_{n}-M\geq p\left(x_{n}-x_{1}\right),\quad M-x_{1}\geq p\left(x_{n}-x_{1}\right)\\ 30\text{ si }x_{i}\leq x_{i}^{\prime},i=1,2,\ldots n,\text{ on }a\\ M\left(x_{1},x_{2},\ldots,x_{n}\right)\leq M\left(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{n}^{\prime}\right)\end{gathered}

From this average of $n$ Numbers $x_{i}$ we deduce another one from $n+1$ numbers by the following method. Let $x_{1}\leq x_{1}\geq\ldots\leq x_{n=1}$ points of ( $a$ , B) and let

x_{n+2,i}^{\prime}=\mathrm{M}\left(x_{1},x_{2},\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1}\right),\quad i=1,2,\ldots,n+1.

Nors

a\leqq x_{1}^{\prime}\leqq x_{2}^{\prime}\leqq\cdots\leqq x_{n+1}^{\prime\prime}\leqq b.

From these numbers, we deduce, by the same method,

u\leqq x_{1}^{\prime\prime}\leqq x_{2}^{\prime\prime}\leqq\cdots\leqq x_{n+1}^{\prime\prime}\leq b

and so on.
Then we have
$\lim x_{1}^{(p)}=\lim x_{2}^{(p)}=\cdots=\lim x_{n+1}^{(p)}=\mathrm{M}^{\prime}\left(x_{1},x_{2},\cdots,x_{n+1}\right)$ For $p\rightarrow\infty$ ,
which is still an average, but of $n+1$ numbers. The author calls it the upper average (Obermittel) of $\mathrm{M}\left(x_{1},x_{2},\ldots,x_{n}\right)$ .

Let us now $M_{i}\left(x_{1},x_{2},\ldots,x_{n}\right),i=1,2,\ldots,m,m$ averages in $(a,b)$ And $\mathrm{N}\left(y_{1},y_{2},\ldots,y_{n}\right)$ an $(\alpha,\beta)$ Or $z=\min f_{\text{, }}\beta=$ max 1. The non-concavity of the function $f$ is then defined by the inequality

\begin{gathered}f\left(M_{1}\left(x_{11},x_{12},\ldots,x_{1n}\right),M_{2}\left(x_{21},x_{22},\ldots,x_{2n}\right),\ldots,M_{m}\left(x_{m1},x_{m2},\ldots,x_{mn}\right)\right)\leqq\\ \leqq\mathrm{N}\left(f\left(x_{11},x_{21},\ldots,x_{m1}\right),\ldots,\quad f\left(x_{1n},x_{2n},\ldots,x_{mn}\right)\right)\end{gathered}

whatever $x_{ij}\in(a,b),i=1,2,\ldots,m,j=1,2,\ldots,n$ G.
Aumann [2 a] then said that $f$ is convex with respect to $\left[\mathrm{M}_{1},\mathrm{M}_{2},\ldots,\mathrm{M}_{m};\mathrm{N}\right]$ and demonstrates that the function is then also convex with respect to $\left[M_{1}^{\prime},M_{2}^{\prime},\ldots,M_{m}^{\prime};N^{\prime}\right]$ , Or $M_{i}^{\prime},N^{\prime}$ are the upper averages of $\mathrm{M}_{i},\mathrm{~N}$ and that, if the function is bounded, it is continuous. By specifying the means that enter into this definition and the function $f$ We obtain several interesting inequalities.
I. Schur [52] considers symmetric functions $/\left(x_{1},x_{2},\ldots,x_{m}\right)$ which verify the inequality

f\left(x_{1},x_{2},\ldots,x_{m}\right)\leqq f\left(\mathbf{X}_{1},\mathbf{X}_{2},\ldots,\mathbf{X}_{m}\right)

(34)

\mathrm{X}_{k}=\sum_{i=1}^{m}a_{ki}x_{i},\quad k=1,2,\ldots,m,\quad a_{ki}\geq 0,\quad\sum_{i=1}^{m}a_{ki}=\sum_{k=1}^{m}a_{ki}=1.

If such a function is differentiable, we have $\left(x_{k}-x_{i}\right)\left(f_{x_{k}}^{\prime}-f_{x_{i}}^{\prime}\right)\leqq 0$ Assuming that $f_{x_{k}}^{\prime}=f_{x_{i}}^{\prime}$ leads $f_{x_{k}}^{\prime\prime}-2f_{x_{k}x_{i}}^{\prime\prime}+f_{x_{i}^{2}}^{\prime\prime}<0$ The equality in (34) is only possible if the $X_{k}$ are a permutation of $x_{i}$ .

This convexity property is verified by the fundamental symmetric functions. $c_{i}=\Sigma x_{1}x_{2}\cdots x_{i}$ , through the functions $\frac{c_{i+1}}{c_{i}}$ , through the functions $\sum_{\begin{subarray}{c}i=1\\ i=1\end{subarray}}^{m}\varphi\left(x_{i}\right)$ , Or $\varphi$ is a non-convex function of order 1 and even by the contractions

\varphi\varphi\left(x_{1}\right)_{\varphi}\left(x_{2}\right)\cdots\varphi\left(x_{i}\right)

if φ is, moreover, non-negative.

BIBLIOGRAPHY

1.

Ascora, G.: a. This is a new algorithm to represent the real-time variable functions. Annali della R. Sc. Norm. Sup. di Pisa (2), 3, 243-253 (1934). b. Only minimal weight is concentrated and the analysis of the functions continues. ibid., (2), 4, 251-266 (1935).
2.

Aumann, G.: a. These functions and induction are unrelated to other conditions. Sitzungsberichte der Bayerischen Akad. Wiss., 11. 3, 403-415 (1933), b. Aufbau son Mittelwerten mehrerer Argumente I. Math. Annalen, 109, 235-253 (1934).
3.

Banacif, S.: On the functional equation $f(x+y)=f(x)+f(y)$ . Fundamenta Math., 1, 123-124 (1920).
4.

Beckenbach, EF: Generalized convex fundions. Amer Bulletin. Math. Goe., 43, 363-371 (1937).
5.

Bennstein, F.: Veber das Cuussche Fehlergesetz. Math. Anualen, 64, 417448 (1907).
6.

Bernstenn, F., Doetsem, G.: Zur Theorie der konoexen Funkionen. Math. Annalen, 76, $514-526$ (1915).
7.

Pernstein, S.: a. Demonstration of Weierstrass's theorem based on the calculus of probabilities. Soc. Math. Charkow, (2), 13, 1-2 (1912). b. Lessons on the extreme properties…, Paris, Gauthier-Villars, X-207 pp. (1926). c. On absolutely monotonic functions. Acta Math., 52, 1-66 (1929).
8.

Blaquier, J.: Sobre dos condiciones curacteristicas de las functiones conperas. Atti Conpresso Bologna, 2, $349-353$ (1930).
9.

Blaschke, W., Prok, G.: Distanschätzungen in Funkionemaum II. Math. Annalen, 77, 277-300 (1916).
10.

Bumere, H.: We conva functions. Amer Transactions. Wath. Soc., 20, 40-44 (1919).
11.

Borel, E.: Let's learn about the functions of real variables. Paris, Gauthier Viilars, VIII-160 pp. (1905).
12.

Bray, 11. E.: On the sewos of a polynomial and of its derivatioes. American Journal of Math., 63, 864-872 (1931).
13.

Colucci, A.: Qualque osservazione sulte funzioni concesse. Bolletino della Un. Mat. J.al. , 7, 139-142 (1928).
14.

Darroux, G. On the composition of fores in statics. Bulletin Se. Math., 9, 281-288 (1875).
15.

Dmi Conaro, A.: Su una disuguaglianza di Jensen. Roudiconti Accad. dei Lincei (6), 17, 1044-1049 (1938).
16.

Favam, J. On average polour. Bulletin Sc. Math., 57, 54-64 (1983).
17.

Frankras, Ph.: Derivatives of higher order as single limits. Amer Bulletin. Math. Soe., 41, 573-582 (1935).
18.

Funwana, M.: Ubber die polynome om der kleisten totalen Schamanong, The Toboku Math. Journal, 3, 129-136 (1913).
19.

Galvant, L.: Sulle funzioni consesse di una o due bariabili definite in un. aggregato qualunque. Rendiconti del Cire. Mast. Palermo, 41, 103-135 (1916).
20.

Hanea, G.: Eine Basis aller Zahlen und die unsteligen Löwagen der Funktionalgletehung $f(x+y)=f(x)-f(y)$ . Math. Annalon, 60, 459-462 (1905).
21.

Hardx, H. II., Lethebworn, J.M., Polya, G.: a. Some simple inequelilies satisfied by concerning finctions. The Messenger of Math., 58, 145-152 (1929). b. Incqualities. Cambridge Univ. Press, XII-314, pp. (1934).
22.

Mösden, O.: Veber emen Miltelwerssars. Nachrichten Cöttingen, 38-4 (1889)).
23.

Hopf. Borlin Dissertation, 30 pp. (1926).
$\int_{a}^{b}\left|f(x)\cdots P_{n}(x)\right|dx$ byl minim. Memoirs Soc. Roy. Bohème, Nr. 15, 25 pp. (1929).
24.

Ingranan, M. H: Solution of hundred fundamental equations relating to a general linear set. Amer Transactions. Math. Soc., 28, 287-300 (1986).
25.

Jageson; D. The theory of approximation. Bitter. Math. Soe Colloquium Publ., VIII-179 pp. (1930),
26.

Jusen, JLWV: On the consecrated functions and the inequalities between the average curves. Acta Math., 30, 175-193 (1906).
27.

Jenswer, B.: a. Bemaerkinger om konodse Kunkioner og Uligheder imellem Middebaerdier I. Mal. Tidsckrift, B, 17-28 (1931). H. Ueber die Verallgeneinerungen des arihmetischen Mittels. Acta Litterarum ac scientiarum, 5, 108-116 (1931).
28.

Kac, M.: A remark on functional equations. Commentarii Math. Helvefici, 9, 170-171 (1937).
29.

Kakeya, S.a. On linear differential inequality. Jownal Soe. physics-math. Tökyö (2), 8, 256261 (1915), b. On some Integral Equations I11. The Töhoka Math. Journal, 8, 14-23 (1925). c. On an inequality betacen the words of in aquation and is deripatized. Proceedings Phys. Math. Soe. Japan (3), 15, 149-154 (1983).
30.

Kanamers, J.: Siir une inequality relative auæ functiona convedes. Math Publications. Univ. Bolgrade, I, 1451148 (1932).
Abis. Konkfor, A., Zobotatere, G.: On a certain minimum. Nonr, Annals of Math. (1), 12, 337-356 (1873).
31.

Knowp, K.: A. Neucre Sätse über Reihen mit postiswem Gliedern Mail. Zoilsehrift, 30, 387-403 (1929). D. Veber die maximaln Abstände mud Verhältnisse perschedener Mittekerte ibid., 39, 768.776 (1935).
32.

Konars, M.: On the functional equation $f(x-t-y)=f(x)+f(y)$ . Bulletia Amer. Math. Soc., 32.689693 (1926).
33.

Krimikos, N.: On multiplely consex or concave functions. Priktika Akad. Athenon, 77, 47-46 (1932).
34.

Leasssume, H.: On ponotual transformations, transforming plans into shapes, which can be defined by analytical processes. Atti della R Accat. Torino, 42, 532-539 (1907).
35.

Manumand, A.: On the derivatives and on the differences of functions of real fine values. Journal de Math., (9), 6, 337-425 (1927).
36.

Markoff, AA On a question posed by Mendeleieff. Budetin Acad. Se de St. Pelesbourg, 62, 1-24 (1890) am russe.
37.

Meisuse, E. J, Mensen's inequality. Butthetin Bitter. Math. Soc. 43, 524527 (1937).
38.

Montsi, P.: A. Sisples approximation polysomes. Buftetin Soc. Matlo.

(1938), third note, 3, 396-402 (1939). S. On bounded solutions and measurable solutions of certain functional equations. Mathematica, 14, 47-106 (1938).
39.

Rado, T.: a. Notes on subharmonic functions. CR Aciad. Sc. Paris, 186, 366-348 (1928) b. We consex functions. Amer Transactions. Math. Soc., 37, 266-285 (1935).
43- Radon, J.: Veber eine Erveiterung des Begriffes der kondeven Funkionen mit einer detwendung auf die Theorie der kompamen Korper. Siloungiberichto K. Akad. Wis. Vienna, 125, 241-258 (191(i).
40.

Rivest, F.: a. On certain simple integral equations. Annales Sr. Ec. Norm. Sup., (3), 28, 33-62 (1911). d. On sub-Turmanic functions and their relation to the potentiality. Acta Math., 48, 329-343 (1926).
41.

Saks, 6.: a. O funkciuch qupuklych i podhurmonicanych. Mathesis Polska, 6. 43 - 65 (1931). h. On the generalized derisatipes. London Math Journal. Soc. 7, 247-251 (1932). c. We subhammonie functions. Acta Litterarmun ac Scientiarum, 5, 187193 (1932).
42.

Scour, I. Ueber cine Klasse von Mittelbildungen nit Anwendungen auf die Demminamuntheomie. Silmungsborichte Borl. Math. Gos., 22, 9-20 (19123)
43.

Stbitaivi, F.: Intosno alle funsioni consesse, Rendiconti Tst. Lombardo (2), 40, 908-919 (1907).
44.

Sebarinske, W. a. Sue l'equation foneoionalnelle $f(x+y)=f(x)+f(y)$ Fundamenta Math., 1, 116-192 (1920). b. On measurable compound functions, ibid., 1, 125-129 (1920). c. On a property of the functions of M. Mamel. ibid., 5, 334-396, (1924).
45.

Stetwuaus, HS Uch die Approximation konoxer perminels lincarep Funktionen. Zeilschrift fiir angelew. Math. und Mechanik, 8, 414-415 (1928).
46.

Stuetwes, Thi J.: Oper ¹ Lagrange's interpolate fomulae. Verslagen en Merledeelingon dor K. Akad. Wolons. Amsterdam (2), 17, 239-254 (1882).
47.

Svon, O.: Grundzüge der Differential-und Integralrehnuni I. Leipzie, BC Tenbmer X-460 pp. (1893).
48.

T&inacini, A. A Acune considerations sul toorma de palor medio. Gionnale di mal. Balfaglini, 51, 66-79 (1913). ©n. Torxorice, P.: a. Sulle fursioni conoesse. Amoli di Mat., (3), 4, 148-151 (1927). b. There are maximum and minimum funds available. Rendiconti Accad. doi Lincei, (6), 14, 172-474 (1931).
49.

VALROM, G.: a. Remarks on inverse functions. Procoedings Phys-Malf. Soc. Tapan, (3), 13, 19-38 (1931). b. Inverse functions and integer functions. Rullotin Soe. Math., 60, 298-287 (1932).
50.

Valleie Poussin, Chi or la: On the convergence of interpolation formulas between equidistant ordinates. Bulletin Acad. Sci. Belgique, 319-410 (1908).
51.

Venbuensky, S.: a. The generalized third deripatipe and its application to the theory of trigonometric series. Proceedings London Math. Soc., (2) 31, 387-406 (1903). H. The generalized fourth derivative. London Math Journal. Soc., 6, 82-84 (1931).
52.

Whitney, H.: Derivatives, difference quotients and Taylor's formula I. Bulletin Amer. Math. Soc., 40, 89-94 (1934).
53.

Winternitz, A.: Veber eine Klasse von linearen Funktionalungleichungen und über konvex Funktionale. Berichte K. Ges. der Wiss. Leipzig, 69, 349-390 (1917).
54.

Young, WH, Young, GG: On the discontinuities of monotone functions of several variables. Proceedings London Math. Soc., (2) 22, 124-142 (1923).

Convex functions

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DOI

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Paper (preprint) in HTML form

SCIENTIFIC AND INDUSTRIAL NEWS

PRESENTATIONS ON THE THEORY OF FUNCTIONS

THE

BY

TIBERIU POPOVICIU

INTRODUCTION

PRELIMINARY CONCEPTS AND NOTATIONS

CHAPTER I

ORDER FUNCTIONS $n$

5.—Definition

CHAPTER II

VARIOUS PROPERTIES OF ORDER FUNCTIONS n

CHAPTER III

GENERALIZATIONS OF ORDER FUNCTIONS $n$

CHAPTER IV

CONVEX HONCTIONS
OF TWO OR MORE VARIABLOS

BIBLIOGRAPHY

Related Posts

Convex functions

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

References

Paper (preprint) in HTML form

SCIENTIFIC AND INDUSTRIAL NEWS

PRESENTATIONS ON THE THEORY OF FUNCTIONS

THE

BY

TIBERIU POPOVICIU

INTRODUCTION

PRELIMINARY CONCEPTS AND NOTATIONS

CHAPTER I

ORDER FUNCTIONSnn

5.—Definition

CHAPTER II

VARIOUS PROPERTIES OF ORDER FUNCTIONS n

CHAPTER III

GENERALIZATIONS OF ORDER FUNCTIONSnn

CHAPTER IV

CONVEX HONCTIONS OF TWO OR MORE VARIABLOS

BIBLIOGRAPHY

Related Posts

ORDER FUNCTIONS $n$

GENERALIZATIONS OF ORDER FUNCTIONS $n$

CONVEX HONCTIONS
OF TWO OR MORE VARIABLOS