- Research
- Open Access
A sharp two-sided inequality for bounding the Wallis ratio
- Senlin Guo^{1}Email author,
- Qi Feng^{2},
- Ya-Qing Bi^{3} and
- Qiu-Ming Luo^{4}
https://doi.org/10.1186/s13660-015-0560-4
© Guo et al.; licensee Springer. 2015
- Received: 12 October 2014
- Accepted: 14 January 2015
- Published: 4 February 2015
Abstract
In this article we establish a sharp two-sided inequality for bounding the Wallis ratio. Some best constants for the estimation of the Wallis ratio are obtained. An asymptotic formula for the Wallis ratio is also presented.
Keywords
- estimation
- inequality
- asymptotic formula
- Wallis ratio
- gamma function
MSC
- 11B65
- 41A44
- 05A10
- 26D20
- 33B15
- 41A60
1 Introduction and main results
In [3], the following result was established:
We also note that in [4] the authors proved the result below.
Our main result may be stated as the following theorem.
Theorem 1
2 Proof of main result
We are now in a position to prove our main result stated in Theorem 1.
Proof of Theorem 1
Declarations
Acknowledgements
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of China under Grant 11401604.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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