Functional inequalities involving numerical differentiation formulas of order two

TL;DR

This paper explores inequalities related to second-order numerical differentiation formulas by expressing them as Stieltjes integrals and applying classical theorems, providing new proofs and extensions for convex functions.

Contribution

It introduces a novel proof of a key inequality for convex functions and extends the results to nonsymmetric cases using integral representations.

Findings

Established a new proof of the inequality for convex functions

Extended the inequality to nonsymmetric cases

Connected numerical differentiation formulas with classical inequalities

Abstract

We write expressions connected with numerical differentiation formulas of order $2$ in the form of Stieltjes integral, then we use Ohlin lemma and Levin-Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality \begin{equation} \label{Dr} f\left(\frac{x+y}{2}\right)\leq\frac{1}{(y-x)^2}\int_x^y\hspace{-2mm}\int_x^yf\left(\frac{s+t}{2}\right)ds\:dt \leq\frac{1}{y-x}\int_x^yf(t)dt \end{equation} satisfied by every convex function $f:\R\to\R$ and we obtain extensions of \rf{Dr}. Then we deal with nonsymmetric inequalities of a similar form.