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

Feng Qi; Bai-Ni Guo

doi:10.3934/cpaa.2009.8.1975

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2009, Volume 8, Issue 6: 1975-1989. Doi: 10.3934/cpaa.2009.8.1975

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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
2.
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010

Received: September 30, 2008

Revised: March 31, 2009

Published: October 2009

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Abstract

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,
logarithmically completely monotonic function,
monotonicity,
convexity,
divided difference,
digamma function,
trigamma function,
ratioof two gamma functions.

Mathematics Subject Classification: Primary: 26A48, 26A51, 33B15; Secondary: 26D20, 65R10.

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