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

TL;DR

The paper investigates complete monotonicity properties of functions built from the poly-double gamma function $\psi_2^{(n)}(x)$, providing integral representations, recurrences, and asymptotics that underlie sharp monotonicity results. By employing Bernstein's theorem and a positivity kernel, it proves that $(-1)^{n+1}\psi_2^{(n)}(x)$ is completely monotone and derives a precise CM condition for composite forms $F_n(x;\omega)$, identifying sharp thresholds $\omega\le\frac{n-2}{n-1}$ and $\omega\ge\frac{n}{n+1}$. It further establishes Turán-type inequalities, sub-/superadditivity properties, and sharp bounds for the ratio $\dfrac{(\psi_2^{(n)}(x))^2}{\psi_2^{(n-1)}(x)\psi_2^{(n+1)}(x)}$, with connections to Hurwitz zeta and polygamma functions. Collectively, the results extend complete monotonicity theory to poly-double gamma functions and illuminate their convexity and inequality structure, offering tools for applications in analysis, number theory, and mathematical physics.

Abstract

In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function \begin{align*} ψ_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 $ψ_2^{(n)}(x)$ and apply these bounds to establish the convexity, subadditivity and superadditivity of $ψ_2^{(n)}(x)$. In the process, various fundamental properties of $ψ_2^{(n)}(x)$ are established, including recurrence relations, integral representations, asymptotic expansions, complete monotonicity, and related inequalities. Graphical illustrations are provided to support the theoretical results.

Figures (6)

• Figure 1: Complete monotonicity of $(-1)^{n+1} \psi_2^{(n)}(x)$ for $n=3$
• Figure 2: Turan-type inequality for $(-1)^{n+1} \psi_2^{(n)}(x)$ for $n=2$
• Figure 3: Graph of $x^{n-1}\psi_2^{(n)}(x)$ for $n=2$ and $x\in(0,40000]$
• Figure 4: Graph of $I_1(a;n)$ for $n=3, 4$ and $a\in(1,2)$
• Figure 5: Complete monotonicity of $F_3(x; 1/4)$

Theorems & Definitions (31)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 2.1
• proof
• Corollary 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3