Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 578310, 11 pages
doi:10.1155/2010/578310
Research Article

An Optimal Double Inequality for Means

Huzhou Broadcast and TV University, Huzhou 313000, China

Received 3 September 2010; Accepted 27 September 2010

Academic Editor: Alberto Cabada

Copyright © 2010 Wei-Mao Qian and Ning-Guo Zheng. 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 𝐿 𝑝 ( π‘Ž , 𝑏 ) , arithmetic mean 𝐴 ( π‘Ž , 𝑏 ) and geometric mean 𝐺 ( π‘Ž , 𝑏 ) of two positive numbers π‘Ž and 𝑏 are defined by 𝐿 𝑝 ( π‘Ž , 𝑏 ) = π‘Ž , π‘Ž = 𝑏 ; 𝐿 𝑝 ( π‘Ž , 𝑏 ) = [ ( π‘Ž 𝑝 + 1 βˆ’ 𝑏 𝑝 + 1 ) / ( ( 𝑝 + 1 ) ( π‘Ž βˆ’ 𝑏 ) ) ] 1 / 𝑝 , 𝑝 β‰  0 , 𝑝 β‰  βˆ’ 1 , π‘Ž β‰  𝑏 ; 𝐿 𝑝 ( π‘Ž , 𝑏 ) = ( 1 / 𝑒 ) ( 𝑏 𝑏 / π‘Ž π‘Ž ) 1 / ( 𝑏 βˆ’ π‘Ž ) , 𝑝 = 0 , π‘Ž β‰  𝑏 ; 𝐿 𝑝 ( π‘Ž , 𝑏 ) = ( 𝑏 βˆ’ π‘Ž ) / ( l n 𝑏 βˆ’ l n π‘Ž ) , 𝑝 = βˆ’ 1 , π‘Ž β‰  𝑏 ; 𝐴 ( π‘Ž , 𝑏 ) = ( π‘Ž + 𝑏 ) / 2 and √ 𝐺 ( π‘Ž , 𝑏 ) = π‘Ž 𝑏 , respectively. In this paper, we give an answer to the open problem: for 𝛼 ∈ ( 0 , 1 ) , what are the greatest value 𝑝 and the least value π‘ž , such that the double inequality 𝐿 𝑝 ( π‘Ž , 𝑏 ) ≀ 𝐺 𝛼 ( π‘Ž , 𝑏 ) 𝐴 1 βˆ’ 𝛼 ( π‘Ž , 𝑏 ) ≀ 𝐿 π‘ž ( π‘Ž , 𝑏 ) holds for all π‘Ž , 𝑏 > 0 ?