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Complete monotonicity of a function involving the gamma function and applications

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Abstract

In the article the authors present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function \(\Gamma (x)\), the \(n\)-th harmonic number \(\sum _{k=1}^n\frac{1}{k}\), and the factorial \(n!\).

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Acknowledgments

The authors would like to express many thanks to the anonymous referee for his/her careful comments on and accurate corrections to the original version of this paper. The present investigation was supported in part by the Natural Science Foundation Project of Chongqing, China under Grant CSTC2011JJA00024, the Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625, and the Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07

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Authors and Affiliations

  1. College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, 028043, Inner Mongolia Autonomous Region, China

    Feng Qi

  2. Department of Mathematics, Chongqing Normal University, Chongqing City, 401331, China

    Qiu-Ming Luo

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  1. Feng Qi
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Correspondence to Feng Qi.

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Qi, F., Luo, QM. Complete monotonicity of a function involving the gamma function and applications. Period Math Hung 69, 159–169 (2014). https://doi.org/10.1007/s10998-014-0056-x

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