Differences of Weighted Mixed Symmetric Means and Related Results

Khuramali Khan,J Pečarić,I Perić

doi:10.1155/2010/289730

Abstract

Some improvements of classical Jensen's inequality are used to define the weighted mixed symmetric means. Exponential convexity and mean value theorems are proved for the differences of these improved inequalities. Related Cauchy means are also defined, and their monotonicity is established as an application.

Highlights

Introduction and Preliminary ResultsFor n ∈ N, let x x1, . . . , xn and p p1, . . . , pn be positive n-tuples such that n i piWe define power means of order r ∈ R, as follows: ⎧ ⎪⎪⎪⎪⎪⎨ n 1/r pixir, r / 0, Mr x, p Mr x1, . . . , xn; p1, . . . , pn i1⎪⎪⎪⎪⎪⎩ Πni 1xipi, r 0.We introduce the mixed symmetric means with positive weights as follows: Journal of Inequalities and Applications Ms1,t x, p; k ⎧⎛ ⎛⎞
Let f be a convex function defined on an interval I ⊆ R, x, p be positive n-tuples such that n i pi
In this paper we prove the exponential convexity of 1.43 for convex functions defined in 1.46 and 1.47 and mean value theorems for the differences given in 1.43

Summary

Introduction and Preliminary Results

For n ∈ N, let x x1, . . . , xn and p p1, . . . , pn be positive n-tuples such that n i pi. Let f be a convex function defined on an interval I ⊆ R, x, p be positive n-tuples such that n i pi. By similar setting in 1.24 , we get the monotonicity of generalized means as follows: Mh3 x, p Mh3,g x, p; 1 ≥ · · · ≥ Mh3,g x, p; k ≥ · · · ≥ Mg3 x, p , 1.30 where f h ◦ g−1 is convex and h is increasing, or f h ◦ g−1 is concave and h is decreasing; Mg3 x, p Mg3,h x, p; 1 ≤ · · · ≤ Mg3,h x, p; k ≤ · · · ≤ Mh3 x, p , 1.31 where f g ◦ h−1 is convex and g is decreasing, or f g ◦ h−1 is concave and g is increasing. In this paper we prove the exponential convexity of 1.43 for convex functions defined in 1.46 and 1.47 and mean value theorems for the differences given in 1.43. We define the corresponding means of Cauchy type and establish their monotonicity

Main Result

Cauchy Mean