We observe that the Hermite-Hadamard inequality written in the form $$f\left(\frac{ x+y}{ 2}\right)\leq\frac{ F(y)-F(x)}{ y-x}\leq\frac{ f(x)+f(y)}{ 2}$$ may be viewed as an inequality between two quadrature operators $f\left(\frac{ x+y}{ 2}\right)$ $\frac{ f(x)+f(y)}{ 2}$ and a differentiation formula $\frac{ F(y)-F(x)}{ y-x}.$ 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 \cite{ Rajba} however to get more interesting result some more general tool must be used. To this end we use Levin-Ste\v{ c}kin theorem which provides necessary and sufficient conditions under which inequalities of the type we consider are satisfied.
Categories: math.CA