On Some Properties and Inequalities for the Nielsen's $\beta$-Function

TL;DR

This paper investigates various mathematical properties and inequalities of Nielsen's eta-function, including convexity, monotonicity, and additivity, contributing to the understanding of this classical special function.

Contribution

It introduces new properties and inequalities for Nielsen's eta-function, enhancing the theoretical framework surrounding this historically significant function.

Findings

Established convexity and monotonicity properties.

Derived new inequalities involving Nielsen's eta-function.

Explored additivity properties of the function.

Abstract

In this study, we obtain some convexity, monotonicity and additivity properties as well as some inequalities involving the Nielsen's $\beta$-function which was introduced in 1906.