# Functional Inequalities Involving Numerical Differentiation Formulas of Order Two

Bulletin of the Malaysian Mathematical Sciences Society, Jan 2017

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{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} satisfied by every convex function $f{:}\,\mathbb R\rightarrow \mathbb R$ and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form.

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Tomasz Szostok. Functional Inequalities Involving Numerical Differentiation Formulas of Order Two, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-14, DOI: 10.1007/s40840-017-0462-3