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Logarithmically completely monotonic functions relating to the gamma function. (English) Zbl 1099.33002
For \(\alpha\in\mathbb{R}\) and \(\beta\geq{0}, \beta\in\mathbb{R}\), define \(f_{\alpha,\beta}(x)=\frac{e^{x}\Gamma(x+\beta)}{x^{x+\beta-\alpha}}\) in \((0,\infty)\). In the paper the following two results are proved.
The function \(f_{\alpha,\beta}(x)\) is logarithmically completely monotonic in \((0,\infty)\) if \(2{\alpha}\leq{1}\leq{\beta}\).
Let \(\alpha\in\mathbb{R}\). The function \(f_{\alpha,1}(x)\) is logarithmically completely monotonic in \((0,\infty)\) if and only if \(2{\alpha}\leq{1}\). So is the function \([f_{\alpha,1}(x)]^{-1}\) if and only if \({\alpha}\geq{1}\).
The proofs use integral expressions for the derivatives of the digamma function, the one-parameter mean of two positive numbers, and an asymptotic expansion for the digamma function. A double inequality for the ratio of two gamma functions, extending a result of G.D. Anderson et.al. appears as corollary. Since some of the authors in the reference list are also \(q\)-experts, we are waiting for the corresponding results for the \(\Gamma_{q}-\)function.

MSC:
33B15 Gamma, beta and polygamma functions
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