Bulletin of the Malaysian Mathematical Sciences Society, Vol. 41, iss. 4 (2018), s. 2053-2066
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 f(x+y2)≤1(y-x)2∫xy∫xyf(s+t2)dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function f:R→R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form.