Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2011; 48(3): 655-667

Printed May 1, 2011

https://doi.org/10.4134/JKMS.2011.48.3.655

Copyright © The Korean Mathematical Society.

A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications

Bai-Ni Guo and Feng Qi

Henan Polytechnic University, Henan Polytechnic University

Abstract

A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

Keywords: completely monotonic function, divided difference, gamma function, psi function, tri-gamma function, probability integral, error function, double factorial, ratio, volume of unit ball, monotonicity, convexity, inequality, generalization, application

MSC numbers: 05A10, 26D15, 26A48, 26A51, 33B15, 33B20