Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 741923, 6 pages
doi:10.1155/2009/741923
Research Article

Two Sharp Inequalities for Power Mean, Geometric Mean, and Harmonic Mean

Yu-Ming Chu1 and Wei-Feng Xia2

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2School of Teacher Education, Huzhou Teachers College, Huzhou 313000, China

Received 23 July 2009; Accepted 30 October 2009

Academic Editor: Wing-Sum Cheung

Copyright © 2009 Yu-Ming Chu and Wei-Feng Xia. 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 p∈R, the power mean of order p of two positive numbers a and b is defined by Mp(a,b)=((ap+bp)/2)1/p,p≠0,  and  Mp(a,b)=ab,   p=0. In this paper, we establish two sharp inequalities as follows: (2/3)G(a,b)+(1/3)H(a,b)⩾M−1/3(a,b) and (1/3)G(a,b)+(2/3)H(a,b)⩾M−2/3(a,b) for all a,b>0. Here G(a,b)=ab and H(a,b)=2ab/(a+b) denote the geometric mean and harmonic mean of a and b, respectively.