On approximately convex and affine functions

Abstract

A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-convex if, for all $x,y\in I$, $t\in[0,1]$ it satisfies $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+t\Phi\big((1-t)|x-y|\big)+(1-t)\Phi\big(t|x-y|\big), $$ where $\Phi:\mathbb{R}_+\to\mathbb{R}_+$ is a nonnegative error function. If $f$ and $-f$ are simultaneously $\Phi$-convex, then $f$ is said to be a $\Phi$-affine function. In the main results of the paper, we describe the structural and inclusion properties of these two classes. We characterize these two classes of functions and investigate their relationship with approximately monotone and approximately-H\"older functions. We also introduce a subclass of error functions which enjoy the so-called $\Gamma$ property and we show that the error function which is the most optimal for a $\Phi$-convex function has to belong to this subclass. The properties of this subclass of error function are investigated as well. Then we offer two formulas for the lower $\Phi$-convex envelop. Besides, a sandwich type theorem is also added.