Logarithmically complete monotonicity of reciprocal arctan function

TL;DR

This paper proves that the reciprocal of the arctangent function is logarithmically completely monotonic on positive real numbers, confirming a conjecture and clarifying its properties related to gamma function ratios.

Contribution

It establishes the logarithmic complete monotonicity of 1/arctan, resolving a conjecture and distinguishing it from being a Stieltjes transform.

Findings

1/ arctan is logarithmically completely monotonic on (0, ∞)

It is not a Stieltjes transform

Confirms a conjecture from prior work

Abstract

We prove the conjecture stated in F. Qi and R. Agarwal, \textit{On complete monotonicity for several classes of functions related to ratios of gamma functions}, J. Inequal. Appl. (2019), 1-42, that the function $1/\arctan$ is logarithmically completely monotonic on $(0,\infty)$, but not a Stieltjes transform.