Schur-m power convexity for a mean of two variables with three parameters

Research Article Schur-Convexity of Two Types of One-Parameter Mean Values in n Variables

and let dμ= du1, . . . ,dun−1 be the differential of the volume in En−1. The weighted arithmetic mean A(x,u) and the power mean Mr(x,u) of order r with respect to the numbers x1,x2, . . . ,xn and the positive weights u1,u2, . . . ,un with ∑n i=1ui = 1 are defined, respectively, as A(x,u) = ∑ni=1uixi, Mr(x,u) = (∑ni=1uixr i ) for r =0, and M0(x,u)= ∏n i=1x ui i . For u=(1/n,1/n, . . . ,1/n), we ...

Schur Convexity of Generalized Heronian Means Involving Two Parameters

The Schur-convexity of the mean of a convex function

The Schur-convexity at the upper and lower limits of the integral for the mean of a convex function is researched. As applications, a form with a parameter of Stolarsky’s mean is obtained and a relevant double inequality that is an extension of a known inequality is established. © 2009 Elsevier Ltd. All rights reserved.

Schur Power Convexity of the Daróczy Means

In this paper, the Schur convexity is generalized to Schur f -convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When f : R+ →R is defined by f (x) = (xm−1)/m if m = 0 and f (x) = lnx if m = 0 , the necessary and sufficient conditions for f -convexity (is called Schur m -power convexity) of Daróczy means are given, which improve, generalize and unify Shi et...

The Schur Convexity for the Generalized Muirhead Mean

For x,y > 0 , a,b ∈ R with a+ b = 0 , the generalized Muirhead mean is defined by M(a,b;x,y) = ( xayb+xbya 2 ) 1 a+b . In this paper, we prove that M(a,b;x,y) is Schur convex with respect to (x,y)∈ (0,∞)×(0,∞) if and only if (a,b)∈ {(a,b)∈R2 : (a−b)2 a+b > 0 & ab 0} and Schur concave with respect to (x,y) ∈ (0,∞)×(0,∞) if and only if (a,b)∈ {(a,b)∈R+ : (a−b)2 a+b & (a,b) = (0,0)}∪{(a,b) ∈ R2 : ...