On the stability of t-convex functions

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, Pales and the author.