On approximately convex and affine functions

A real valued function $f$ defined on a 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 $\Phi:\mathbb{R}_+\to\mathbb{R}_+$ nonnegative error function. If and $-f$ are simultaneously $\Phi$-convex, then said to be $\Phi$-affine In the main results of paper, we describe structural inclusion properties these two classes. We characterize classes functions investigate their relationship with approximately monotone approximately-Holder functions. also introduce subclass which enjoy so-called $\Gamma$ property show that most optimal has belong this subclass. The investigated as well. Then offer formulas lower envelop. Besides, sandwich type theorem added.