The aim of this paper is to prove a regularity theorem for real valued subquadratic mappings that are solutions of the inequality \[ \varphi (x + y) + \varphi(x - y) \leq 2\varphi(x) + 2\varphi(y),\quad x, y \in X, where \(X = (X, +)\) is a topological group.
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.
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