Further refinement of results about mixed symmetric means and Cauchy means

Abstract

Recently, Horvath introduced a new method to refine
the well known discrete Jensen’s inequality (see [2]). He also gave a parameter dependant refinement of the discrete Jensen’s inequality (see [3]). We apply the new exponential convexity method as illustrated in [7], to the functionals obtained from the refinement results of [2] and [3]. In this way we are able to generalize the results given in [4] as well as given in [1].