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

TL;DR

This paper investigates the completely monotonic degree of a function involving derivatives of the digamma function, providing proofs, conjectures, and discussing related properties like logarithmic concavity.

Contribution

It establishes the completely monotonic degree of a specific function involving the digamma function as 4 and explores related monotonicity and concavity properties.

Findings

The completely monotonic degree of $[ ext{psi}'(x)]^2 + ext{psi}''(x)$ is 4.

A function $f(x)$ is strongly completely monotonic iff $xf(x)$ is completely monotonic.

The paper conjectures the degree of a related function and discusses properties of logarithmic concavity.

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 $[\psi'(x)]^2+\psi''(x)$ 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 $[\psi'(x)]^2+\psi''(x)$, presents the logarithmic concavity and monotonicity of an elementary function, and poses an open problem on convolution of logarithmically concave functions.