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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MATHEMATICAL ANALYSIS. - On convex functions of a real variable. Note (^(2){ }^{2}) by MT Popovici.
I. We consider uniform, real functions of the real variablexx, defined on a linear and bounded set E. Let
By definition, the functionf(x)f(x)is convex, not concave, polynomial, ^((1)){ }^{(1)}We will show this in more detail in a later paper. (^(2))\left({ }^{2}\right)Meeting of June 1, 1930.
CR, 19.30, 1st Semester. (T. 190, N•25.)
concave or non-concex of ordernndepending on the expression
East> , >= , < , <= 0>, \geq, <, \leq 0, for any set ofn+2n+2pointsx_(1),x_(2),dots,x_(n+2)x_{1}, x_{2}, \ldots, x_{n+2}belonging to E. All these functions form the class of order functionsnn. Iff(x)f(x)is convex or non-concave, the function -f(x)f(x)is concave or non-convex of the same order.
An order functionn > 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 ordernncannot take moren+1n+1times the same value. A convex and bounded function of ordernn, defined on a dense set in an interval, reaches its maximum in[(n+3)/(2)]\left[\frac{n+3}{2}\right]points and its minimum in[(n+2)/(2)]\left[\frac{n+2}{2}\right]points at most;[alpha];[\alpha]denoting the largest integer not greater thanalpha\alphaII .
An order functionn > 1n>1, defined and bounded in an open interval(a,b)(a, b), has a derivative throughout this interval. If the function is convex of ordernn, this derivative is convex of ordern-1n-1, and vice versa. In all cases, it is a function of ordern-1n-1It follows that the derivative(n-1)^("lìme ")(n-1)^{\text {lìme }}ìexists.
The derivative of a function of a given order is bounded on any interval completely inside (a,ba, b).
Iff^((n+1))(x)f^{(n+1)}(x)exists, the conditionf^((n+1))(x) > 0f^{(n+1)}(x)>0is sufficient so thatf(x)f(x)either convex of ordernnthe conditionj^((n+1))(x) >= 0j^{(n+1)}(x) \geq 0is necessary and sufficient for thatf(x)f(x)either non-concave of ordernn.
Iff(x)f(x)is of ordernnIn (a,ba, b) And
|f(x)| < A,quad" dans "(a,b),|f(x)|<\mathrm{A}, \quad \text { dans }(a, b),
for the pointsxxlocated in the interval {(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\}, c,c_(1)c, c_{1}being two independent constants ofnn
For the other points of the interval (a,ba, b), the derivative generally has a
growth rate that depends on the function, as demonstrated by the example of polynomials (').
III. The functionf(x)f(x), of a determined convexity character, extends over a setE_(1)\mathrm{E}_{1}if we can find a function of the same nature defined onE+E_(1)\mathrm{E}+\mathrm{E}_{1}which coincides withf(x)f(x)on E.
The extension of the functionf(x)f(x)of ordern > 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,ba, bLet a polynomial present the convexity characteristics of the function. These approximation polynomials are, for example, the polynomials of M. S. polytein:
IV. Let's take any number of points in the interval(a,b)(a, b)
a <= x_(1) < x_(2) < dots < x_(m) <= ba \leq x_{1}<x_{2}<\ldots<x_{m} \leq b
and a functionf(x)f(x)defined in (a,ba, b
If the set of quantities
is limited, we can say thatf(x)f(x)is a function itn^("ieme ")n^{\text {ieme }}bounded variation (^(2){ }^{2}).
Ifn > 0n>0, the function is continuous. Ifn > 1n>1The function has a derivative. This derivative is at(n-1)^("ieme ")(n-1)^{\text {ieme }}bounded variation. We then deduce that any function withn^("iémo ")n^{\text {iémo }}ébounded variation is the sum of a non-concave function and a non-convex function of ordernn.
(^(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))\left(^{2}\right)The casen=0n=0is classic. The casen=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].
