On Conditions for the Levin-Stečkin inequality and applications

Abstract

Using an identity involving the Green function, we give a generalization of the Levin-Stečkin inequality as a result concerning difference of integral arithmetic means with different measures. The results are given with usual conditions for the Levin-Stečkin type inequalities. As a special case, generalization of the Levin-Stečkin inequality is given. We also prove that result for general measures can be easily reduced to one
single condition. Comparing the two approaches we get some surprisingly good estimations.