A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

- Journal title : Journal of the Korean Mathematical Society
- Volume 48, Issue 3, 2011, pp.655-667
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2011.48.3.655

Title & Authors

A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

Guo, Bai-Ni; Qi, Feng;

Guo, Bai-Ni; Qi, Feng;

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 and 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;

Language

English

Cited by

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