Generalizations of Sherman’s inequality

Abstract

The concept of majorization is a powerful and useful tool which arises frequently in many different areas of research. Together with the concept of Schur-convexity it gives an important characterization of convex functions. The well known Majorization theorem plays a very important role in majorization theory—it gives a relation between one-dimensional convex functions and n-dimensional Schur-convex functions. A more general result was obtained by S. Sherman. In this paper, we get generalizations of these results for n-convex functions using Taylor’s interpolating polynomial and the Čebyšev functional. We apply the exponentially convex method in order to interpret our results in the form of exponentially, and in the special case logarithmically convex functions. The outcome is some new classes of two-parameter Cauchy-type means.