A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

Abstract

A class of functions involving the divided differences of the psi function and the polygamma 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 ratio of two gamma functions and originating from 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.

11 pages

References in corpus (3)

• A property of logarithmically absolutely monotonic functions and the logarithmically complete monotonicity of a power-exponential function
• Some logarithmically completely monotonic functions related to the gamma function
• An alternative proof of Elezović-Giordano-Pečarić's theorem

Cited by in corpus (9)

• Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem
• Refinements of lower bounds for polygamma functions
• A logarithmically completely monotonic function involving the ratio of gamma functions
• Some best approximation formulas and inequalities for Wallis ratio
• Completely monotonic degree of a function involving the tri- and tetra-gamma functions
• Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic
• A completely monotonic function involving the tri- and tetra-gamma functions
• A completely monotonic function involving the gamma and tri-gamma functions
• Some inequalities for the trigamma function in terms of the digamma function