Integral representations and properties of some functions involving the logarithmic function

TL;DR

This paper derives integral representations for complex functions involving the logarithm, explores their properties like operator monotonicity and Bernstein functions, and confirms a conjecture on their complete monotonicity.

Contribution

It introduces new integral representations and properties of functions involving the logarithm, including proving a conjecture on their complete monotonicity.

Findings

Established integral representations using Cauchy formula

Identified functions as operator monotone, Bernstein, and Stieltjes functions

Verified a conjecture on complete monotonicity

Abstract

By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such as being operator monotone function, being complete Bernstein function, and being Stieltjes function, for these functions, and verify a conjecture on complete monotonicity of a function involving the logarithmic function.