Journal of Applied Mathematics
Volume 2012 (2012), Article ID 471096, 14 pages
http://dx.doi.org/10.1155/2012/471096
Research Article

Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

Wei-Mao Qian1 and Zhong-Hua Shen2

1School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
2Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Received 10 October 2011; Accepted 1 December 2011

Academic Editor: Md. Sazzad Chowdhury

Copyright © 2012 Wei-Mao Qian and Zhong-Hua Shen. 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 prove that 𝛼 𝐻 ( 𝑎 , 𝑏 ) + ( 1 − 𝛼 ) 𝐿 ( 𝑎 , 𝑏 ) > 𝑀 ( 1 − 4 𝛼 ) / 3 ( 𝑎 , 𝑏 ) for 𝛼 ∈ ( 0 , 1 ) and all 𝑎 , 𝑏 > 0 with 𝑎 ≠ 𝑏 if and only if 𝛼 ∈ [ 1 / 4 , 1 ) and 𝛼 𝐻 ( 𝑎 , 𝑏 ) + ( 1 − 𝛼 ) 𝐿 ( 𝑎 , 𝑏 ) < 𝑀 ( 1 − 4 𝛼 ) / 3 ( 𝑎 , 𝑏 ) if and only if √ 𝛼 ∈ ( 0 , 3 3 4 5 / 8 0 − 1 1 / 1 6 ) , and the parameter ( 1 − 4 𝛼 ) / 3 is the best possible in either case. Here, 𝐻 ( 𝑎 , 𝑏 ) = 2 𝑎 𝑏 / ( 𝑎 + 𝑏 ) , 𝐿 ( 𝑎 , 𝑏 ) = ( 𝑎 − 𝑏 ) / ( l o g 𝑎 − l o g 𝑏 ) , and 𝑀 𝑝 ( 𝑎 , 𝑏 ) = ( ( 𝑎 𝑝 + 𝑏 𝑝 ) / 2 ) 1 / 𝑝 ( 𝑝 ≠ 0 ) and 𝑀 0 √ ( 𝑎 , 𝑏 ) = 𝑎 𝑏 are the harmonic, logarithmic, and pth power means of a and b, respectively.