Estimates for approximately Jensen convex functions

Abstract

In this paper functions \(f \colon D \to\mathbb{R}\) satisfying the inequality

are studied, where \(D\) is a nonempty convex subset of a real linear space \(X\) and \(\varphi \colon \{\frac12(x-y) : x,y \in D\}\to\mathbb{R}\) is a so-called error function. In this situation \(f\) is said to be \(\varphi\)-Jensen convex. The main results show that for all \(\varphi\)-Jensen convex function \(f \colon D \to\mathbb{R}\), for all rational \(\lambda\in[0,1]\)and \(x,y\in D\), the following inequality holds

The infinite series on the right hand side is always convergent, moreover, for all rational \(\lambda\in[0,1]\), it can be evaluated as a finite sum.