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A note on complete monotonicity of the remainder in stirling’s formula

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An Erratum to this article was published on 01 July 2015

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Abstract

Mortici [C. Mortici, “On the monotonicity and convexity of the remainder of the Stirling formula,” Appl. Math. Lett. 24 (6), 869–871 (2011)] showed that the function −x −1 θ‴(x), where θ(x) is given by

$\Gamma (x + 1) = \sqrt {2\pi } \left( {\frac{x} {e}} \right)^x e^{\theta (x)/12x} = \sqrt {2\pi x} \left( {\frac{x} {e}} \right)^x e^{\sigma (x)/12x} $

is strictly completely monotonic on (0,∞). The aim of this paper is to prove that σ‴(x) is strictly completely monotonic on (0,∞) by using the theory of Laplace transforms.

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Authors and Affiliations

  1. School of Mathematics and Physics, University of Science and Technology, Beijing, China

    S. Guo

  2. School of Mathematics and Statistics, Hainan Normal University, Haikou, China

    Y. Shen

  3. School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, China

    X. Li

Authors
  1. S. Guo
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  2. Y. Shen
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  3. X. Li
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Correspondence to S. Guo.

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The article was submitted by the authors for the English version of the journal.

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Guo, S., Shen, Y. & Li, X. A note on complete monotonicity of the remainder in stirling’s formula. Math Notes 97, 961–964 (2015). https://doi.org/10.1134/S0001434615050326

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