Journal of Applied Mathematics
Volume 2011 (2011), Article ID 618929, 9 pages
http://dx.doi.org/10.1155/2011/618929
Research Article

Optimal Inequalities between Harmonic, Geometric, Logarithmic, and Arithmetic-Geometric Means

Yu-Ming Chu and Miao-Kun Wang

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

Received 19 July 2011; Accepted 4 September 2011

Academic Editor: Laurent Gosse

Copyright © 2011 Yu-Ming Chu and Miao-Kun Wang. 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

We find the least values 𝑝 , 𝑞 , and 𝑠 in (0, 1/2) such that the inequalities 𝐻 ( 𝑝 𝑎 + ( 1 − 𝑝 ) 𝑏 , 𝑝 𝑏 + ( 1 − 𝑝 ) 𝑎 ) > A G ( 𝑎 , 𝑏 ) , 𝐺 ( 𝑞 𝑎 + ( 1 − 𝑞 ) 𝑏 , 𝑞 𝑏 + ( 1 − 𝑞 ) 𝑎 ) > A G ( 𝑎 , 𝑏 ) , and 𝐿 ( 𝑠 𝑎 + ( 1 − 𝑠 ) 𝑏 , 𝑠 𝑏 + ( 1 − 𝑠 ) 𝑎 ) > A G ( 𝑎 , 𝑏 ) hold for all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 , respectively. Here A G ( 𝑎 , 𝑏 ) , 𝐻 ( 𝑎 , 𝑏 ) , 𝐺 ( 𝑎 , 𝑏 ) , and 𝐿 ( 𝑎 , 𝑏 ) denote the arithmetic-geometric, harmonic, geometric, and logarithmic means of two positive numbers 𝑎 and 𝑏 , respectively.