NECESSARY AND SUFFICIENT CONDITIONS FOR A DIFFERENCE DEFINED BY FOUR DERIVATIVES OF A FUNCTION CONTAINING TRIGAMMA FUNCTION TO BE COMPLETELY MONOTONIC

In this paper, using convolution theorem for Laplaces transforms, logarithmic convexity of the gamma function, Bernsteins theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Using the Cebysev integral inequality, the author also presents logarithmic convexity of the sequence of polygamma functions.

Automated summary

This paper investigates the complete monotonicity of a difference defined by four derivatives of a function containing trigamma function, and presents the necessary and sufficient conditions as well as the proof of logarithmic convexity by applying various techniques.