Some Comparison Inequalities for Generalized Muirhead and Identric Means

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)