Complete Monotonicity of the function involving derivatives of Barnes G-function

TL;DR

This paper investigates the complete monotonicity and related properties of functions involving derivatives of the Barnes G-function, providing bounds, inequalities, and applications to convexity and subadditivity.

Contribution

It establishes the complete monotonicity of functions involving derivatives of the Barnes G-function and derives new bounds, inequalities, and properties such as convexity and subadditivity.

Findings

Proves complete monotonicity of -derivatives of the Barnes G-function

Derives bounds and inequalities for -derivatives of the Barnes G-function

Establishes convexity, subadditivity, and superadditivity properties

Abstract

In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function \begin{align*} \psi_2^{(n)}(x) = (-1)^{n+1} n! \sum_{k=0}^{\infty} \dfrac{(1+k)}{(x+k)^{n+1}}, \quad x > 0, \ n\geq 2. \end{align*} Consequently, we derive bounds for the ratio involving $\psi_2^{(n)}(x)$ and apply these bounds to establish the convexity, subadditivity and superadditivity of $\psi_2^{(n)}(x)$. In the process, various fundamental properties of $\psi_2^{(n)}(x)$ are established, including recurrence relations, integral representations, asymptotic expansions, complete monotonicity, and related inequalities. Graphical illustrations are provided to support the theoretical results.