Complete monotonicity of a function involving derivatives of the Barnes G-function

In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function align* ψ₂^ (n) (x) = (-1) ^n+1 n! ₊=₀^ (1+k) (x+k) ^{n+1}, x > 0, \ n 2. align* 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.