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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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##### References:
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