On some conversions of the Jensen-Steffensen inequality

Some conversions of the Jensen-Steffensen inequality for convex functions are considered. Applying exp-convex method improvements and reverses of the Slater-Pečarić inequality are obtained. Related Cauchy’s type means are defined and some basic properties are given

Extensions and improvements of Sherman’s and related inequalities for n-convex functions

This paper gives extensions and improvements of Sherman’s inequality for n-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established

Cauchy type means related to the converse Jensen-Steffensen inequality

In this paper we apply so called exp-convex method to the converse Jensen-Steffensen inequality in order to interpret it in the form of exponentially convex functions. The outcome is a new class of Cauchy type means and some new interesting inequalities related to them

On an upper bound for Sherman’s inequality

Abstract Considering a weighted relation of majorization, Sherman obtained a useful generalization of the classical majorization inequality. The aim of this paper is to extend Sherman’s inequality to convex functions of higher order. An upper bound for Sherman’s inequality, as well as for generalized Sherman’s inequality, is given with some applications. As easy consequences, some new bounds for Jensen’s inequality can be derived

Generalizations of Sherman’s inequality by Lidstone’s interpolating polynomial

Abstract In majorization theory, the well-known majorization theorem plays a very important role. A more general result was obtained by Sherman. In this paper, concerning 2n-convex functions, we get generalizations of these results applying Lidstone’s interpolating polynomials and the Čebyšev functional. Using the obtained results, we generate a new family of exponentially convex functions. The results are some new classes of two-parameter Cauchy type means