Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 392431, 11 pages
doi:10.1155/2010/392431
Research Article

A Class of Logarithmically Completely Monotonic Functions Associated with a Gamma Function

Tie-Hong Zhao and Yu-Ming Chu

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

Received 15 November 2010; Accepted 27 December 2010

Academic Editor: Shusen Ding

Copyright © 2010 Tie-Hong Zhao 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

We show that the function 𝐺 𝛼 , 𝛽 ( 𝑥 ) = 𝑒 𝑥 Γ ( 𝑥 + 𝑎 ) / ( 𝑥 + 𝛽 ) 𝑥 + 𝛽 is strictly logarithmically completely monotonic on ( 0 , ∞ ) if and only if ( 𝛼 , 𝛽 ) ∈ { ( 𝛼 , 𝛽 ) ∶ 0 < 𝛼 ≤ 𝛽 } and [ 𝐺 𝛼 , 𝛽 ( 𝑥 ) ] − 1 is strictly logarithmically completely monotonic on ( 0 , ∞ ) if and only if ( 𝛼 , 𝛽 ) ∈ { ( 𝛼 , 𝛽 ) ∶ 0 < 𝛽 ≤ 𝛼 − 1 / 2 } .