Completely monotonic degree of a function involving the tri- and tetra-gamma functions

Abstract

Let $\psi (x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $\Gamma (x)$. In the paper, the author shows that the completely monotonic degree of the function $[{\psi }^{\prime }(x){]}^2+{\psi }^{\prime \prime }(x)$ is $4$, surveys the history and motivation of the topic, supplies a proof for the claim that a function $f(x)$ is strongly completely monotonic if and only if the function $xf(x)$ is completely monotonic, conjectures the completely monotonic degree of a function involving $[{\psi }^{\prime }(x){]}^2+{\psi }^{\prime \prime }(x)$, presents the logarithmic concavity and monotonicity of an elementary function, and poses an open problem on convolution of logarithmically concave functions.

29 pages

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Cited by in corpus (7)

• Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function
• Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function
• Some inequalities for the trigamma function in terms of the digamma function
• A double inequality for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function
• An analytic generalization of the Catalan numbers and its integral representation
• Partial solutions to several conjectures on completely monotonic degrees for remainders in asymptotic expansions of the digamma function
• Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark--Ismail's two conjectures