Inequalities of the Hermite-Hadamard Type Involving Numerical Differentiation Formulas

摘要

We observe that the Hermite-Hadamard inequality written in the form f (x + y/2) <= F(y) - F(x)/y - x <= f(x) + f(y)/2 may be viewed as an inequality between two quadrature operators f (x+y/2) f(x)+ f(y)/2 and a differentiation formula F(y)-F(x)/y-x. 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-Steckin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied.