Mercer and Wu–Srivastava generalisations of Steffensen’s inequality

摘要

In this paper generalisations of Steffensen’s inequality obtained by Pečarić, Mercer and Wu–Srivastava are related. Some of this generalisations are showed to be equivalent and using Wu–Srivastava refinement of Steffensen’s inequality refined versions of Pečarić and Mercer’s results are obtained. Using Wu–Srivastava sharpened and generalized version of Mercer’s result sharpened version of generalization given by Pečarić is obtained. Furthermore, weaker conditions for some generalisations and refinements are obtained. Moreover, functionals defined as the difference between the left-hand and the right-hand side of generalizations and refinements of Steffensen’s inequality are studied. New Stolarsky type means, using several families of exponentially convex functions, related to defined functionals are generated.