Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 806825, 10 pages
doi:10.1155/2010/806825
Research Article

Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means

Bo-Yong Long1,2 and Yu-Ming Chu3

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2School of Mathematical Sciences, Anhui University, Hefei 230039, China
3Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 20 October 2009; Revised 16 December 2009; Accepted 1 February 2010

Academic Editor: Alexander I. Domoshnitsky

Copyright © 2010 Bo-Yong Long and Yu-Ming Chu. 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(a,b), arithmetic mean A(a,b), and geometric mean G(a,b) of two positive numbers a and b are defined by Lp(a,b)=a, for a=b, Lp(a,b)=[(bp+1-ap+1)/((p+1)(b-a))]1/p, for p≠0, p≠-1, and a≠b, Lp(a,b)=(1/e)(bb/aa)1/(b-a), for p=0, and a≠b, Lp(a,b)=(b-a)/(log⁡⁡b-log⁡⁡a), for p=-1, and a≠b, A(a,b)=(a+b)/2, and G(a,b)=ab, respectively. In this paper, we find the greatest value p (or least value q, resp.) such that the inequality Lp(a,b)<αA(a,b)+(1-α)G(a,b) (or αA(a,b)+(1-α)G(a,b)<Lq(a,b), resp.) holds for α∈(0,1/2)(or α∈(1/2,1), resp.) and all a,b>0 with a≠b.