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Integral representation and functional inequalities involving generalized polylogarithm

Deepshikha Mishra, A. Swaminathan

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.

Integral representation and functional inequalities involving generalized polylogarithm

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 86 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Organization
  4. preliminaries
  5. Generalized polylogarithm as a single integral
  6. Generalized polylogarithm as a double integral

Key Result

Theorem 2.1

Schilling_2021_bernstein function(Bernstein theorem) Let $f: (0,\infty) \mapsto \mathbb{R}$ be a completely monotonic function. Then it can be expressed as the Laplace transform of a unique measure $\mu$ defined on the interval $[0,\infty)$. In other words, for every $x > 0$, Conversely, if $\mathscr{L}(\mu; x)< \infty$ for every $x>0$, then the function $x \mapsto \mathscr{L}{(\mu; x)}$ is compl

Figures (5)

  • Figure 1: Complete monotonicity of $\psi(p)$
  • Figure 2: Bounds for generalized polylogarithm $\Phi_{p, q}(a, b; x)$.
  • Figure 3: Convexity of $\log \psi(p)$.
  • Figure 4: Turan inequality for $\psi(p)$.
  • Figure 5: Turan inequality for the $\Phi_{p,q}(a,b;x)$ with respect to $q$

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Remark 2.1
  • Theorem 3.1
  • ...and 17 more