Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 295620, 10 pages
Research Article

Some Comparison Inequalities for Generalized Muirhead and Identric Means

Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 21 December 2009; Accepted 23 January 2010

Academic Editor: Jong Kim

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.


For x,y>0, a,b, with a+b0, 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/(yx), xy, 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 xy and (a,b){(a,b)2:a+b>0,ab0,2(ab)23(a+b)+10,3(ab)22(a+b)0}; (2) M(a,b;x,y)<I(x,y) for all x,y>0 with xy and (a,b){(a,b)2:a0,b0,3(ab)22(a+b)0}{(a,b)2:a+b<0}; (3) if (a,b){(a,b)2:a>0,b>0,3(ab)22(a+b)>0}{(a,b)2:ab<0,3(ab)22(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).