Generalization and sharpness of the power means inequality and their applications

Abstract

In this paper, we generalize and sharpen the power means inequality by using the theory of majorization and the analytic techniques. Our results unify some optimal versions of the power means inequality. As application, a well-known conjectured inequality proposed by Janous et al. is proven. Furthermore, these results are used for studying a class of geometric inequalities for simplex, from which, some interesting inequalities including the refined Euler inequality and the reversed FinslerHadwiger type inequality are obtained.  2005 Elsevier Inc. All rights reserved.

A relation of weak majorization and its applications to certain inequalities for means

“…P r o o f. Since E k (x α ) and E * k (x α ) are Schur-concave on R n ++ , then the majorization relation (3) with the definition of Schur-concavity yield inequalities (16) and (17). (16) and (17) we get the following corollary.…”

“…A good survey on the theory of majorization was given by Marshall and Olkin in [4]. Recently, the authors have given considerable attention to the applications of majorization in the field of inequalities, for details, we refer the reader to our papers [2], [3], [5]- [14] and [16]- [19].…”

Schur convexity of Bonferroni harmonic mean

“…As supplements to the Schur convexity of functions, the Schur geometrically convex functions and Schur harmonically convex functions were investigated [8,21,26,27].…”

General norm inequalities for bounded linear operators in Hilber spaces

“…Using [Theorem 2, [10]] and applying the same manner used in proving the above theorems, we derive the following result.…”