On the convex functions of a real variable

Tiberiu Popoviciu
(original), convexity, Mathematical Analysis, paper

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T. Popoviciu

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T. Popoviciu, Sur les fonctions convexes d’une variable réelle, Comptes Rendus de l’Academie des Sciences de Paris, 190 (1930), pp. 1481-1483 (in French).

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Comptes Rendus de l’Academie des Sciences de Paris

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Académie des Sciences, Paris; Elsevier, Paris.

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[JFM 56.0241.03]

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1930 -Popoviciu- Proceedings - On convex functions of a real variable
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MATHEMATICAL ANALYSIS. - On convex functions of a real variable. Note ( 2 2 ^(2){ }^{2}2) by MT Popovici.
I. We consider uniform, real functions of the real variable x x xxx, defined on a linear and bounded set E. Let
U ( α 1 , α 2 , … , α k ; f ) = | 1 α 1 α 1 2 … α 1 k − 2 f ( α 2 ) 1 α 2 α 2 2 … α 2 k − 2 f ( α 2 ) . . . . . … … . … . . 1 α k α k 2 … α k k − 2 f ( α k ) | , V ( α 1 , α 2 , … , α k ) = | 1 α 1 α 1 2 … α 2 k − 2 1 α 2 α 2 2 … α 2 k − 2 . . . . … … . 1 α k α k 2 … α k k − 1 | . U α 1 , α 2 , … , α k ; f = 1 α 1 α 1 2 … α 1 k − 2 f α 2 1 α 2 α 2 2 … α 2 k − 2 f α 2 . . . . . … … . … . . 1 α k α k 2 … α k k − 2 f α k , V α 1 , α 2 , … , α k = 1 α 1 α 1 2 … α 2 k − 2 1 α 2 α 2 2 … α 2 k − 2 . . . . … … . 1 α k α k 2 … α k k − 1 . {:[U(alpha_(1),alpha_(2),dots,alpha_(k);f)=|[1,alpha_(1),alpha_(1)^(2),dots,alpha_(1)^(k-2),f(alpha_(2))],[1,alpha_(2) ,alpha_(2)^(2),dots,alpha_(2)^(k-2),f(alpha_(2))],[.,..,..,dots,dots.,dots..],[1,alpha_(k),alpha_(k)^(2),dots,alpha_(k) ^(k-2),f(alpha_(k))]|","],[V(alpha_(1),alpha_(2),dots,alpha_(k))=|[1,alpha_(1),alpha_(1)^(2),dots,alpha_(2)^(k-2)],[1, alpha_(2),alpha_(2)^(2),dots,alpha_(2)^(k-2)],[.,..,.,dots,dots.],[1,alpha_(k),alpha_(k)^(2),dots,alpha_(k)^(k-1)]|.]:}\begin{aligned} & \mathrm{U}\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k} ; f\right)=\left|\begin{array}{cccccc} 1 & \alpha_{1} & \alpha_{1}^{2} & \ldots & \alpha_{1}^{k-2} & f\left(\alpha_{2}\right) \\ 1 & \alpha_{2} & \alpha_{2}^{2} & \ldots & \alpha_{2}^{k-2} & f\left(\alpha_{2}\right) \\ . & . . & . . & \ldots & \ldots . & \ldots . . \\ 1 & \alpha_{k} & \alpha_{k}^{2} & \ldots & \alpha_{k}^{k-2} & f\left(\alpha_{k}\right) \end{array}\right|, \\ & \mathrm{V}\left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{k}\right)=\left|\begin{array}{ccccc} 1 & \alpha_{1} & \alpha_{1}^{2} & \ldots & \alpha_{2}^{k-2} \\ 1 & \alpha_{2} & \alpha_{2}^{2} & \ldots & \alpha_{2}^{k-2} \\ . & . . & . & \ldots & \ldots . \\ 1 & \alpha_{k} & \alpha_{k}^{2} & \ldots & \alpha_{k}^{k-1} \end{array}\right| . \end{aligned}U(α1,α2,…,αk;f)=|1α1α12…α1k−2f(α2)1α2α22…α2k−2f(α2).....…….…..1αkαk2…αkk−2f(αk)|,V(α1,α2,…,αk)=|1α1α12…α2k−21α2α22…α2k−2....…….1αkαk2…αkk−1|.
By definition, the function f ( x ) f ( x ) f(x)f(x)f(x)is convex, not concave, polynomial,
( 1 ) ( 1 ) ^((1)){ }^{(1)}(1)We will show this in more detail in a later paper.
( 2 ) 2 (^(2))\left({ }^{2}\right)(2)Meeting of June 1, 1930.
CR, 19.30, 1st Semester. (T. 190, N•25.)

concave or non-concex of order n n nnndepending on the expression

U ( x 1 , x 2 , … x n ÷ 2 ; f ) V ( x 1 , x 2 , … , x n ÷ 2 ) U x 1 , x 2 , … x n ÷ 2 ; f V x 1 , x 2 , … , x n ÷ 2 (U(x_(1),x_(2),dotsx_(n-:2);f))/(V(x_(1),x_(2),dots,x_(n-:2)))\frac{\mathrm{U}\left(x_{1}, x_{2}, \ldots x_{n \div 2}; f\right)}{\mathrm{V}\left(x_{1}, x_{2}, \ldots, x_{n \div 2}\right)}U(x1,x2,…xn÷2;f)V(x1,x2,…,xn÷2)
East > , ≥ , < , ≤ 0 > , ≥ , < , ≤ 0 > , >= , < , <= 0>, \geq, <, \leq 0>,≥,<,≤0, for any set of n + 2 n + 2 n+2n+2n+2points x 1 , x 2 , … , x n + 2 x 1 , x 2 , … , x n + 2 x_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}x1,x2,…,xn+2belonging to E. All these functions form the class of order functions n n nnn. If f ( x ) f ( x ) f(x)f(x)f(x)is convex or non-concave, the function - f ( x ) f ( x ) f(x)f(x)f(x)is concave or non-convex of the same order.
An order function n > 0 n > 0 n > 0n>0n>0, defined and bounded on a closed set, is continuous, except perhaps at the ends.
A function of a given order, defined on a closed set, reaches its maximum and its minimum.
A convex or concave function of order n n nnncannot take more n + 1 n + 1 n+1n+1n+1times the same value. A convex and bounded function of order n n nnn, defined on a dense set in an interval, reaches its maximum in [ n + 3 2 ] n + 3 2 [(n+3)/(2)]\left[\frac{n+3}{2}\right][n+32]points and its minimum in [ n + 2 2 ] n + 2 2 [(n+2)/(2)]\left[\frac{n+2}{2}\right][n+22]points at most ; [ α ] ; [ α ] ;[alpha];[\alpha];[α]denoting the largest integer not greater than α α alpha\alphaαII .
An order function n > 1 n > 1 n > 1n>1n>1, defined and bounded in an open interval ( a , b ) ( a , b ) (a,b)(a, b)(has,b), has a derivative throughout this interval. If the function is convex of order n n nnn, this derivative is convex of order n − 1 n − 1 n-1n-1n−1, and vice versa. In all cases, it is a function of order n − 1 n − 1 n-1n-1n−1It follows that the derivative ( n − 1 ) lìme ( n − 1 ) lìme  (n-1)^("lìme ")(n-1)^{\text {lìme }}(n−1)lime exists.
The derivative of a function of a given order is bounded on any interval completely inside ( a , b a , b a,ba, bhas,b).
If f ( n + 1 ) ( x ) f ( n + 1 ) ( x ) f^((n+1))(x)f^{(n+1)}(x)f(n+1)(x)exists, the condition f ( n + 1 ) ( x ) > 0 f ( n + 1 ) ( x ) > 0 f^((n+1))(x) > 0f^{(n+1)}(x)>0f(n+1)(x)>0is sufficient so that f ( x ) f ( x ) f(x)f(x)f(x)either convex of order n n nnnthe condition j ( n + 1 ) ( x ) ≥ 0 j ( n + 1 ) ( x ) ≥ 0 j^((n+1))(x) >= 0j^{(n+1)}(x) \geq 0j(n+1)(x)≥0is necessary and sufficient for that f ( x ) f ( x ) f(x)f(x)f(x)either non-concave of order n n nnn.
If f ( x ) f ( x ) f(x)f(x)f(x)is of order n n nnnIn ( a , b a , b a,ba, bhas,b) And
| f ( x ) | < A , dans ( a , b ) , | f ( x ) | < A ,  dans  ( a , b ) , |f(x)| < A,quad" dans "(a,b),|f(x)|<\mathrm{A}, \quad \text { dans }(a, b),|f(x)|<HAS, In (has,b),
we have
| f ′ ( x ) | < c n 2 b − a A , | f ′ ( x ) ( x − a ) ( b − x ) | < c 1 n A , f ′ ( x ) < c n 2 b − a A , f ′ ( x ) ( x − a ) ( b − x ) < c 1 n A , |f^(')(x)| < c(n^(2))/(b-a)A,quad|f^(')(x)sqrt((x-a)(b-x))| < c_(1)nA,\left|f^{\prime}(x)\right|<c \frac{n^{2}}{b-a} \mathrm{~A}, \quad\left|f^{\prime}(x) \sqrt{(x-a)(b-x)}\right|<c_{1} n \mathrm{~A},|f′(x)|<cn2b−has HAS,|f′(x)(x−has)(b−x)|<c1n HAS,
for the points x x xxxlocated in the interval
{ a ( cos ⁡ π n + cos ⁡ π 3 n ) + b ( 1 − cos ⁡ π 2 n ) 1 + cos ⁡ π n , b ( cos ⁡ π n + cos ⁡ π 2 n ) ÷ a ( 1 − cos ⁡ π 2 n ) 1 + cos ⁡ π n } a cos ⁡ π n + cos ⁡ π 3 n + b 1 − cos ⁡ π 2 n 1 + cos ⁡ π n , b cos ⁡ π n + cos ⁡ π 2 n ÷ a 1 − cos ⁡ π 2 n 1 + cos ⁡ π n {(a(cos((pi )/(n))+cos((pi)/(3n)))+b(1-cos((pi)/(2n))))/(1+cos((pi )/(n))),(b(cos((pi )/(n))+cos((pi)/(2n)))-:a(1-cos((pi)/(2n))))/(1+cos((pi )/(n)))}\left\{\frac{a\left(\cos \frac{\pi}{n}+\cos \frac{\pi}{3 n}\right)+b\left(1-\cos \frac{\pi}{2 n}\right)}{1+\cos \frac{\pi}{n}}, \frac{b\left(\cos \frac{\pi}{n}+\cos \frac{\pi}{2 n}\right) \div a\left(1-\cos \frac{\pi}{2 n}\right)}{1+\cos \frac{\pi}{n}}\right\}{has(cos⁡πn+cos⁡π3n)+b(1−cos⁡π2n)1+cos⁡πn,b(cos⁡πn+cos⁡π2n)÷has(1−cos⁡π2n)1+cos⁡πn},
c , c 1 c , c 1 c,c_(1)c, c_{1}c,c1being two independent constants of n n nnn
For the other points of the interval ( a , b a , b a,ba, bhas,b), the derivative generally has a
growth rate that depends on the function, as demonstrated by the example of polynomials (').
III. The function f ( x ) f ( x ) f(x)f(x)f(x), of a determined convexity character, extends over a set E 1 E 1 E_(1)\mathrm{E}_{1}E1if we can find a function of the same nature defined on E + E 1 E + E 1 E+E_(1)\mathrm{E}+\mathrm{E}_{1}E+E1which coincides with f ( x ) f ( x ) f(x)f(x)f(x)on E.
The extension of the function f ( x ) f ( x ) f(x)f(x)f(x)of order n > 1 n > 1 n > 1n>1n>1Extending a function defined and bounded on a finite set of points is generally impossible. Conversely, if the order is 0 or i, the extension is always possible. A convex (or concave) function of order 0 or i defined on a finite set extends by a polynomial of the same nature. This is a consequence of the fact that a continuous function in ( a , b a , b a,ba, bhas,bLet a polynomial present the convexity characteristics of the function. These approximation polynomials are, for example, the polynomials of M. S. polytein:
∑ i = 0 n f ( a + i b − a n ) ( n i ) ( x − a ) n ( b − a ) n − i ( b − a ) n . ∑ i = 0 n   f a + i b − a n ( n i ) ( x − a ) n ( b − a ) n − i ( b − a ) n . (sum_(i=0)^(n)f(a+i(b-a)/(n))((n)/(i))(x-a)^(n)(b-a)^(n-i))/((b-a)^(n)).\frac{\sum_{i=0}^{n} f\left(a+i \frac{b-a}{n}\right)\binom{n}{i}(x-a)^{n}(b-a)^{n-i}}{(b-a)^{n}} .∑i=0nf(has+ib−hasn)(ni)(x−has)n(b−has)n−i(b−has)n.
IV. Let's take any number of points in the interval ( a , b ) ( a , b ) (a,b)(a, b)(has,b)
a ≤ x 1 < x 2 < … < x m ≤ b a ≤ x 1 < x 2 < … < x m ≤ b a <= x_(1) < x_(2) < dots < x_(m) <= ba \leq x_{1}<x_{2}<\ldots<x_{m} \leq bhas≤x1<x2<…<xm≤b
and a function f ( x ) f ( x ) f(x)f(x)f(x)defined in ( a , b a , b a,ba, bhas,b
If the set of quantities

is limited, we can say that f ( x ) f ( x ) f(x)f(x)f(x)is a function it n ieme n ieme  n^("ieme ")n^{\text {ieme }}nth bounded variation ( 2 2 ^(2){ }^{2}2).
If n > 0 n > 0 n > 0n>0n>0, the function is continuous. If n > 1 n > 1 n > 1n>1n>1The function has a derivative. This derivative is at ( n − 1 ) ieme ( n − 1 ) ieme  (n-1)^("ieme ")(n-1)^{\text {ieme }}(n−1)th bounded variation. We then deduce that any function with n iémo n iémo  n^("iémo ")n^{\text {iémo }}niémo bounded variation is the sum of a non-concave function and a non-convex function of order n n nnn.
  1. ( 1 1 ^(1){ }^{1}1) Particular convex polynomials have already been studied. See W. Bünčka and J. Geronimus, Ueber die monotone Polynome welche die minimal Abweichung von Null haben (Math. Zeitschrift, 30, 1929, p. 35-).
    ( 2 ) 2 (^(2))\left(^{2}\right)(2)The case n = 0 n = 0 n=0n=0n=0is classic. The case n = 1 n = 1 n=1n=1n=1was studied by MA Wintennitz, Ueber eine Klasse von linearen unktional-Ungleichungen und über Konvexe Funktionale [Berichte der Süchsischen ries. der Wissens. su Leipsig (math. phys. Klasse), 69, 1917, p. 346].
1930

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