Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

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 fx+y2≤1(y-x)2∫xy∫xyfs+t2dsdt≤1y-x∫xyf(t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f\left( \frac{x+y}{2}\right) \le \frac{1}{(y-x)^2} \int _x^y\int _x^yf\left( \frac{s+t}{2}\right) \hbox {d}s\,\hbox {d}t \le \frac{1}{y-x}\int _x^yf(t)\hbox {d}t \end{aligned}$$\end{document}satisfied by every convex function f:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f{:}\,\mathbb R\rightarrow \mathbb R$$\end{document} and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form.