Baire measurability of \((M,N)\)-Wright convex functions

Michał Lewicki

Commentationes Mathematicae, Tome 48 (2008) no. 1, pp. 75-83.

Résumé

Let \(I \subset \mathbb{R}\) be an open interval and \(M, N \colon I^2 \to I\) be means on \(I\). Let \(\varphi\colon I \to \mathbb{R}\) be a solution of the functional equation \[ \varphi(M (x, y)) + \varphi(N (x, y)) = \varphi(x) + \varphi(y),\quad x, y \in I \] We give sufficient conditions on \(M, N\) and the function \(\varphi\) such that for every Baire measurable solution \(f \colon I \to \mathbb{R}\) of the functional inequality \[ f (M (x, y)) + f (N (x, y)) \leq f (x) + f (y),\quad x, y \in I, \] the function \(f \circ \varphi^{-1} \colon \varphi(I) \to \mathbb{R}\) is convex.

Détail

DOI : 10.14708/cm.v48i1.5260

Mots-clés : Wright convexity, functional inequalities, regularity property, Baire measurable functions, continuous functions