Complete Monotonicity of the function involving derivatives of Barnes G-function
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911258442203136 |
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| author | Mishra, Deepshikha Swaminathan, A. |
| author_facet | Mishra, Deepshikha Swaminathan, A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07444 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete Monotonicity of the function involving derivatives of Barnes G-function Mishra, Deepshikha Swaminathan, A. General Mathematics 33B15, 26D07, 26A48 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. |
| title | Complete Monotonicity of the function involving derivatives of Barnes G-function |
| topic | General Mathematics 33B15, 26D07, 26A48 |
| url | https://arxiv.org/abs/2511.07444 |