Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 763252, 7 pages
doi:10.1155/2009/763252
Research Article

Inequalities for Generalized Logarithmic Means

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 2 June 2009; Accepted 10 December 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 generalized logarithmic mean Lp of two positive numbers a and b is defined as Lp(a,b)=a, for a=b, LP(a,b)=[(bp+1−ap+1)/(p+1)(b−a)]1/p , for a≠b, p≠−1, p≠0, LP(a,b)=(b−a)/(log⁡b−log⁡a), for a≠b, p=−1, and LP(a,b)=(1/e)(bb/aa)1/(b−a) , for a≠b, p=0. In this paper, we prove that G(a,b)+H(a,b)⩾2L−7/2(a,b),A(a,b)+H(a,b)⩾2L−2(a,b), and L−5(a,b)⩾H(a,b) for all a,b>0, and the constants −7/2,−2, and −5 cannot be improved for the corresponding inequalities. Here A(a,b)=(a+b)/2=L1(a,b),G(a,b)=ab=L−2(a,b), and H(a,b)=2ab/(a+b) denote the arithmetic, geometric, and harmonic means of a and b, respectively.