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 \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.