Inequalities of the Hermite-Hadamard type involving numerical differentiation formulas

TL;DR

This paper explores inequalities related to the Hermite-Hadamard inequality, extending it to more complex forms involving numerical differentiation formulas and employing advanced theorems to establish necessary and sufficient conditions.

Contribution

It introduces generalized inequalities of Hermite-Hadamard type involving numerical differentiation formulas and applies Levin-Stečkin theorem for establishing conditions.

Findings

Extended Hermite-Hadamard inequalities with complex terms

Utilized Levin-Stečkin theorem for inequality conditions

Compared results with previous approaches like Ohlin lemma

Abstract

We observe that the Hermite-Hadamard inequality written in the form $$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$ may be viewed as an inequality between two quadrature operators $f\left(\frac{x+y}{2}\right)$ $\frac{f(x)+f(y)}{2}$ and a differentiation formula $\frac{F(y)-F(x)}{y-x}.$ We extend this inequality, replacing the middle term by more complicated ones. As it turns out in some cases it suffices to use Ohlin lemma as it was done in a recent paper \cite{Rajba} however to get more interesting result some more general tool must be used. To this end we use Levin-Ste\v{c}kin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied.