2009•arXiv (Cornell University)Open access

Necessary and sufficient conditions for a function involving divided differences of the di- and tri-gamma functions to be completely monotonic

Feng Qi, Bai‐Ni Guo

Abstract

In the present paper, necessary and sufficient conditions are established for a function involving divided differences of the digamma and trigamma functions to be completely monotonic. Consequently, necessary and sufficient conditions are derived for a function involving the ratio of two gamma functions to be logarithmically completely monotonic, and some double inequalities are deduced for bounding divided differences of polygamma functions.

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Key concepts: Monotonic function, Digamma function, Gamma function, Bounding overwatch, Mathematics, Function (biology), Pure mathematics, Applied mathematics