A Class of logarithmically completely monotonic functions relating the $q$-gamma function and applications

TL;DR

This paper investigates the logarithmically complete monotonicity of functions involving the q-gamma function for q in (0,1), deriving new inequalities and estimates related to Stirling's formula remainders.

Contribution

It introduces new monotonicity properties for q-gamma functions and derives novel inequalities and bounds for Stirling's approximation remainders.

Findings

Established logarithmic complete monotonicity for q-gamma related functions.

Derived new inequalities for the q-gamma function.

Provided improved estimates for Stirling's formula remainders.

Abstract

In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are established. Furthermore, let the sequence $r_n$ be defined by $$n!=\sqrt{2\pi n}(n/e)^n e^{r_n}.$$ We establish new estimates for Stirling's formula remainder $r_n.$