Completely monotonic function associated with the Gamma functions and proof of Wallis' inequality

Authors

  • Chao-Ping Chen
  • Feng Qi

DOI:

https://doi.org/10.5556/j.tkjm.36.2005.101

Abstract

We prove: (i) A logarithmically completely monotonic function is completely monotonic. (ii) For $ x>0 $ and $ n=0, 1, 2, \ldots $, then

$$ (-1)^{n}\left(\ln \frac{x \Gamma(x)}{\sqrt{x+1/4}\,\Gamma(x+1/2)}\right)^{(n)}>0. $$

(iii) For all natural numbers $ n $, then

$$ \frac1{\sqrt{\pi(n+4/ \pi-1)}}\leq \frac{(2n-1)!!}{(2n)!!}<\frac1{\sqrt{\pi(n+1/4)}}. $$

The constants $ \frac{4}{\pi}-1$ and $\frac14 $ are the best possible.

Author Biographies

Chao-Ping Chen

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China.

Feng Qi

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, China.

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Published

2005-12-31

How to Cite

Chen, C.-P., & Qi, F. (2005). Completely monotonic function associated with the Gamma functions and proof of Wallis’ inequality. Tamkang Journal of Mathematics, 36(4), 303-307. https://doi.org/10.5556/j.tkjm.36.2005.101

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Section

Papers

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