Estimates for approximately Jensen convex functions

Abstract

In this paper functions $f:D\to \mathbb{R}$ satisfying the inequality \[ f\Big(\frac{x+y}{2}\Big)\leq\frac12f(x)+\frac12f(y) +φ\Big(\frac{x-y}{2}\Big) \qquad(x,y\in D) \] are studied, where $D$ is a nonempty convex subset of a real linear space $X$ and $\phi :\{\frac{1}{2}(x-y):x,y\in D\}\to \mathbb{R}$ is a so-called error function. In this situation $f$ is said to be $\phi$-Jensen convex. The main results show that for all $\phi$-Jensen convex function $f:D\to \mathbb{R}$, for all rational $\lambda \in [0,1]$ and $x,y\in D$, the following inequality holds \[ f(λx+(1-λ)y) \leq λf(x)+(1-λ)f(y)+\sum_{k=0}^\infty \frac{1}{2^k}φ\big(\mbox{dist}(2^kλ,\mathbb{Z})\cdot(x-y)\big). \] 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.