Zur Theorie der konvexen Funktionen

Abstract

Under einer konvexen FunkCion vers~eh~ man eine in einem Intervall (a, b) Ftir jeden Weft definler~ eindeufige reelle Fnnl~ion f(x), die fftr je zwei Werte x tund x~ im ]n~rvall (a, b) der Bedingung gen~g~. J. L. W. V. Jensen*) hat gezeig~: Ist die kanvexe Funktion i~ dem Interva~l (a, b) nach oben beschr5nkt, so ist sie dort stetig. (Die S~etigkeit erstreckt sich iibrigens nut auf die inneren Punkte des In~ervalls, an den Enden brauch~ die Funktion nicht stetig zu sein, z. B. f(x) = x ~ fiir-l<x<+l, =2 ,, x=-i *) J. L. W. V. Jansen, Sat le~ fonv~ions e~nvexes et les in~galitds entre leurs valeurs moyaunes.

Convex functions

“…This result has been extended by Bernstein and Doetsch [9], Blumberg [ll], Sierpenski [67], and Ostrowski [49].…”

Convexity with respect to families of means

“…On the other hand, for q < 0, we have that f p,q ≤ 0 on I p . Therefore, by the Bernstein-Doetsch theorem [3], f p,q is convex. The interval I p is open, hence f p,q is locally Lipschitz on I p .…”

“…Consequently, by Sierpiński's generalization [13] of the Bernstein-Doetsch Theorem [3], for every p ∈ P , f p,q(p) is continuous and hence it is also convex.…”

“…Jensen [7], [8] verified that continuous Jensen convex functions are convex. Bernstein and Doetsch [3] showed that upper boundedness on a nonempty open set and Jensen convexity imply convexity. Sierpiński [13] proved that upper boundedness on a set of positive Lebesgue measure and Jensen convexity are also sufficient conditions for convexity.…”

Midconvex functions in locally compact groups

“…Furthermore he showed that f is continuous providing it has finite lim sup at each point, and especially if it is assumed to be upper semi-continuous or to be bounded from above. Many authors have been then interested in the problem of continuity of functions satisfying Jensen's functional inequality 2f (x) ≤ f (x + h) + f(x − h) (1) where f : Ω → R is a real function defined on a convex subset Ω of a real vector space X on which a suitable topology is given, and the variables x and h are restricted only to the conditions x, x + h, x − h ∈ Ω. These functions are also called convex in [11], convex, convex(J), J-convex or midconvex in the literature.…”