Copyright © 2010 Miao-Kun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For x,y>0,  a,b∈ℝ, with a+b≠0, the generalized Muirhead mean M(a,b;x,y) with parameters a and b and the identric mean I(x,y) are defined by M(a,b;x,y)=((xayb+xbya)/2)1/(a+b) and I(x,y)=(1/e)(yy/xx)1/(y−x), x≠y, I(x,y)=x, x=y, respectively. In this paper, the following results are established: (1) M(a,b;x,y)>I(x,y) for all x,y>0 with x≠y and (a,b)∈{(a,b)∈ℝ2:a+b>0, ab≤0, 2(a−b)2−3(a+b)+1≥0, 3(a−b)2−2(a+b)≥0}; (2) M(a,b;x,y)<I(x,y) for all x,y>0 with x≠y and (a,b)∈{(a,b)∈ℝ2:a≥0, b≥0, 3(a−b)2−2(a+b)≤0}∪{(a,b)∈ℝ2:a+b<0}; (3) if (a,b)∈{(a,b)∈ℝ2:a>0, b>0, 3(a−b)2−2(a+b)>0}∪{(a,b)∈ℝ2:ab<0, 3(a−b)2−2(a+b)<0}, then there exist x1,y1,x2,y2>0 such that M(a,b;x1,y1)>I(x1,y1) and M(a,b;x2,y2)<I(x2,y2).