Abstract and Applied Analysis
Volume 2011 (2011), Article ID 896483, 13 pages
http://dx.doi.org/10.1155/2011/896483
Research Article

Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

Tie-Hong Zhao,1 Yu-Ming Chu,1 and Hua Wang2

1Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Changsha University of Science and Technology, Changsha 410076, China

Received 26 March 2011; Accepted 18 May 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Tie-Hong Zhao et al. 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 the function 𝑓 𝛼 , 𝛽 ( 𝑥 ) = Γ 𝛽 ( 𝑥 + 𝛼 ) / 𝑥 𝛼 Γ ( 𝛽 𝑥 ) is strictly logarithmically completely monotonic on ( 0 , ∞ ) if √ ( 𝛼 , 𝛽 ) ∈ { ( 𝛼 , 𝛽 ) ∶ 1 / 𝛼 ≤ 𝛽 ≤ 1 , 𝛼 ≠ 1 } ∪ { ( 𝛼 , 𝛽 ) ∶ 0 < 𝛽 ≤ 1 , 𝜑 1 ( 𝛼 , 𝛽 ) ≥ 0 , 𝜑 2 ( 𝛼 , 𝛽 ) ≥ 0 } and [ 𝑓 𝛼 , 𝛽 ( 𝑥 ) ] − 1 is strictly logarithmically completely monotonic on ( 0 , ∞ ) if √ ( 𝛼 , 𝛽 ) ∈ { ( 𝛼 , 𝛽 ) ∶ 0 < 𝛼 ≤ 1 / 2 , 0 < 𝛽 ≤ 1 } ∪ { ( 𝛼 , 𝛽 ) ∶ 1 ≤ 𝛽 ≤ 1 / √ 𝛼 ≤ 2 , 𝛼 ≠ 1 } ∪ { ( 𝛼 , 𝛽 ) ∶ 1 / 2 ≤ 𝛼 < 1 , 𝛽 ≥ 1 / ( 1 − 𝛼 ) } , where 𝜑 1 ( 𝛼 , 𝛽 ) = ( 𝛼 2 + 𝛼 − 1 ) 𝛽 2 + ( 2 𝛼 2 − 3 𝛼 + 1 ) 𝛽 − 𝛼 and 𝜑 2 ( 𝛼 , 𝛽 ) = ( 𝛼 − 1 ) 𝛽 2 + ( 2 𝛼 2 − 5 𝛼 + 2 ) 𝛽 − 1 .