Inequalities of the Hermite–Hadamard Type Involving Numerical Differentiation Formulas

We observe that the Hermite–Hadamard inequality written in the form fx+y2≤F(y)-F(x)y-x≤f(x)+f(y)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\left(\frac{x+y}{2}\right)\leq\frac{F(y)-F(x)}{y-x}\leq\frac{f(x)+f(y)}{2}$$\end{document}may be viewed as an inequality between two quadrature operators fx+y2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f\left(\frac{x+y}{2}\right)}$$\end{document}f(x)+f(y)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{f(x)+f(y)}{2}}$$\end{document} and a differentiation formula F(y)-F(x)y-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{F(y)-F(x)}{y-x}}$$\end{document}. 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 (Rajba, Math Inequal Appl 17(2):557–571, 2014) however to get more interesting result some more general tool must be used. To this end we use Levin–Stečkin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied.