Integral representation and functional inequalities involving generalized polylogarithm

TL;DR

This paper derives new integral representations of the generalized polylogarithm using Dirichlet series and Hadamard convolution, and explores its properties like monotonicity, convexity, and inequalities.

Contribution

It introduces two novel integral representations of the generalized polylogarithm and establishes its key properties, including bounds and inequalities.

Findings

Derived single and double integral representations of the generalized polylogarithm.

Established properties such as complete monotonicity, Turan inequality, and convexity.

Provided an alternative proof of an existing integral representation.

Abstract

The purpose of this manuscript is to derive two distinct integral representations of the generalized polylogarithm using two different techniques. The first approach involves the Dirichlet series and its Laplace representation, which leads to a single integral representation. The second approach utilizes the Hadamard convolution, resulting in a double integral representation. As a consequence, an integral representation of the Lerch transcendent function is obtained. Furthermore, we establish properties such as complete monotonicity, Turan inequality, convexity, and bounds of the generalized polylogarithm. Finally, we provide an alternative proof of an existing integral representation of the generalized polylogarithm using the Hadamard convolution.