Some properties of extended remainder of Binet's first formula for logarithm of gamma function

TL;DR

This paper extends Binet's first formula for the gamma function's logarithm and explores its properties, including inequalities and monotonicity, contributing to a deeper understanding of special functions.

Contribution

It introduces an extension of Binet's first formula and analyzes its properties, such as inequalities, star-shapedness, sub-additivity, and complete monotonicity.

Findings

Derived new inequalities for the extended remainder

Established star-shaped and sub-additive properties

Proved complete monotonicity of the extended remainder

Abstract

In the paper, we extend Binet's first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet's first formula for the logarithm of the gamma function and related functions.