On the stability of t-convex functions

Summary.

A real-valued function f defined on an open convex set $D\subseteq X$ is called (d, t)-convex if it satisfies

$$f(tx + (1 - t)y)\leq tf(x)+(1 - t)f(y) + d(x, y)$$

for all $x,\, y \in D$, where $d : X {\times} X \rightarrow{\mathbb{R}}$ is a given function and t $\in$]0, 1[ is a fixed parameter.

The main result of the paper states that if f is locally bounded from above at a point of D and (d, t)-convex then it satisfies the convexity-type inequality (under some assumptions)

$$f(sx + (1 - s)y)\leq sf(x) + (1 - s)f(y) +\varphi(s)d(x, y)$$

for all $x, y \in D$ and s $\in$ [0, 1], where $\varphi : [0, 1] \rightarrow {\mathbb{R}}$ is defined as the fixed point of a certain contraction. The main result of this paper offers a generalization of the celebrated Bernstein and Doetsch theorem and the recent results by Nikodem and Ng, Páles and the author.