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

2025/11/03 by Mishra, Deepshikha, Swaminathan, A.
#26A48 #26D07 #33B15 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2511.07444

Abstract

In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function ψ₂⁽ⁿ⁾(x) = (-1)ⁿ⁺¹ n! ∑_k=0^∞ \dfrac(1+k)(x+k)ⁿ⁺¹, x gt; 0, n≥ 2. Consequently, we derive bounds for the ratio involving ψ₂⁽ⁿ⁾(x) and apply these bounds to establish the convexity, subadditivity and superadditivity of ψ₂⁽ⁿ⁾(x). In the process, various fundamental properties of ψ₂⁽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.

Citations

• Inequalities for 1/(1-cos(x)) and its derivatives
• A complete Bernstein function related to the fractal dimension of Pascal's pyramid modulo a prime
• Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function
• The barnes G function and its relations with sums and products of generalized Gamma convolution variables

Related

• Complete monotonicity properties of a function involving the polygamma function
• Complete Monotonicity and Inequalities of Functions Involving Γ-function
• Monotonicity properties related to the ratio of two gamma functions
• Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function
• Monotonicity, convexity, and inequalities for functions involving gamma function