Logarithmically complete monotonicity properties relating to the gamma function (Q638143)

Summary: We prove that the function \(f_{\alpha,\beta}(x) = \Gamma^{\beta}(x + \alpha) / x^{\alpha}{\Gamma}({\beta}x)\) is strictly logarithmically completely monotonic on \((0, \infty)\) if \((\alpha, \beta) \in \{(\alpha, \beta) : 1/\root \of{\alpha} \leq {\beta} \leq 1, {\alpha} \neq 1\} \cup \{(\alpha, \beta) : 0 < {\beta} \leq 1, \varphi_1 (\alpha, \beta) \geq 0, \varphi_2 (\alpha, \beta) \geq 0\}\) and \([f_{\alpha,\beta}(x)]^{-1}\) is strictly logarithmically completely monotonic on \((0, \infty)\) if \((\alpha, \beta) \in \{(\alpha, \beta) : 0 < {\alpha} \leq 1/2, 0 < {\beta} \leq 1\} \cup \{(\alpha, \beta) : 1 \leq {\beta} \leq 1/\root \of{\alpha} \leq \root \of{2}, {\alpha} \neq 1\} \cup \{(\alpha, \beta) : 1/2 \leq {\alpha} < 1, {\beta} \geq 1/(1 - \alpha)\}\), where \(\varphi_1 (\alpha, \beta) = (\alpha^2 + \alpha - 1){\beta}^2 + (2{\alpha}^2 - 3{\alpha} + 1){\beta} - {\alpha}\) and \(\varphi_2(\alpha, \beta) = (\alpha - 1){\beta}^2 + (2\alpha^2 - 5\alpha + 2){\beta} - 1\).