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Complete Monotonicities of Functions Involving the Gamma and Digamma Functions
Feng Qi,Bai-Ni Guo +1 more
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In this article, the completely monotonic results of the functions [Γ(x+ 1)]1/x, [ Γ (x+ α+1)]1/(x+α) [ γ(x + 1)] 1/x, [γ(ex+ 1)α and [α+ 1] 1/ex xα in x ∈ (−1,∞) for α ∈ R are obtained.Abstract:
In the article, the completely monotonic results of the functions [Γ(x+ 1)]1/x, [Γ(x+α+1)]1/(x+α) [Γ(x+1)]1/x , [Γ(x+1)]1/x (x+1)α and [Γ(x+1)]1/x xα in x ∈ (−1,∞) for α ∈ R are obtained. In the final, three open problems are posed.read more
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Harmonic Analysis and the Theory of Probability. By S. Bochner pp. 176. 35s. 1955. (California University Press and Cambridge University Press)
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Bounds for the ratio of two gamma functions.
TL;DR: A long history of bounding the ratio for and, various origins of this topic are clarified, several developed courses are followed, different results are compared, useful methods are summarized, new advances are presented, some related problems are pointed out, and related references are collected as discussed by the authors.
Bounds for the ratio of two gamma functions--From Wendel's limit to Elezović-Giordano-Pečarić's theorem
TL;DR: A long history of bounding the ratio for and, various origins of this topic are clarified, several developed courses are followed, different results are compared, useful methods are summarized, new advances are presented, some related problems are pointed out, and related references are collected as mentioned in this paper.
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Integral representation of some functions related to the Gamma function
TL;DR: In this paper, it was shown that the functions Θ(Θ(x) = [Gamma (x + 1)]^{1/x} (1 + 1/x)^x /x) and Θ (Θ (x, Θ) = Θ((Θ + 1)/x)) are Stieltjes transforms.
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Completely monotonic functions involving the gamma and q-gamma functions
TL;DR: In this paper, an infinite family of functions involving the gamma function whose logarithmic derivatives are completely monotonic were given. And they were shown to give an infinitely divisible probability distribution for specific combinations of the gamma and q-gamma functions.
References
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Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints
K. S. Kölbig,F. Schäff +1 more