Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

Abstract

In this paper, the connection between the functional inequalities $$f\left(\frac{x+y}{2}\right)\le \frac{f(x)+f(y)}{2}+{\alpha }_J(x-y)(x,y\in D)$$ and $${\int }_0^{1}f(tx+(1-t)y)\rho (t)dt\le \lambda f(x)+(1-\lambda )f(y)+{\alpha }_H(x-y)(x,y\in D)$$ is investigated, where $D$ is a convex subset of a linear space, $f:D\to \mathbb{R}$, ${\alpha }_H,{\alpha }_J:D-D\to \mathbb{R}$ are even functions, $\lambda \in [0,1]$, and $\rho :[0,1]\to {\mathbb{R}}_{+}$ is an integrable nonnegative function with ${\int }_0^1\rho (t)dt=1$.