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November  2009, 8(6): 1975-1989. doi: 10.3934/cpaa.2009.8.1975

Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications

Feng Qi 1, and  Bai-Ni Guo 2,

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang City, Henan Province, 453007, China
2. 

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010

Received  October 2008 Revised  April 2009 Published  August 2009

In the present paper, we establish the complete monotonicity of two functions involving divided differences of the digamma function $\psi$ and the trigamma function $\psi'$. Applying these monotonicity, we provide an alternative proof for the monotonicity and convexity of a function derived from bounding the ratio of two gamma functions, procure the logarithmically completely monotonic property of a function involving the ratio of two gamma functions, and obtain new bounds for the ratio of two gamma functions.
Keywords: Completely monotonic function, divided difference, digamma function, logarithmically completely monotonic function, convexity, trigamma function, monotonicity, ratioof two gamma functions..
Mathematics Subject Classification: Primary: 26A48, 26A51, 33B15; Secondary: 26D20, 65R10. .
Citation: Feng Qi, Bai-Ni Guo. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1975-1989. doi: 10.3934/cpaa.2009.8.1975
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