Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

Abstract

In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function

“…Remark 7. This paper is a revised version of the electronic preprint [6] and the third one in a series of articles including [7,9,10,11,12,13,14,15,18].…”

Necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic

“…and J 0,µ0 (x) = H µ0 (x). In the paper [5], the author established the following necessary and sufficient conditions:…”

Decreasing properties of two ratios defined by three and four polygamma functions

“…Remark 23. This paper is a revised version of the electronic preprint [15] and is the eighth one in a series of articles including [17,18,[20][21][22][23][24][25]29].…”