# American Institute of Mathematical Sciences

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

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