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 functionMSC
11B65 41A44 05A10 26D20 33B15 41A601 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
References
- Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1970) Google Scholar
- Mortici, C: Completely monotone functions and the Wallis ratio. Appl. Math. Lett. 25, 717-722 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Zhao, Y, Wu, Q: Wallis inequality with a parameter. J. Inequal. Pure Appl. Math. 7, 56 (2006) MathSciNetGoogle Scholar
- Guo, B-N, Qi, F: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J. Korean Math. Soc. 48, 655-667 (2011) View ArticleMATHMathSciNetGoogle Scholar
- Gradshteyn, IS, Ryzhik, IM: Table of Integrals, Series, and Products, 6th edn. Academic Press, New York (2000) MATHGoogle Scholar
- Andrews, GE, Askey, R, Roy, R: Special Functions. Cambridge University Press, Cambridge (1999) View ArticleMATHGoogle Scholar
- Guo, S: Monotonicity and concavity properties of some functions involving the gamma function with applications. J. Inequal. Pure Appl. Math. 7, 45 (2006) Google Scholar
- Guo, S: Logarithmically completely monotonic functions and applications. Appl. Math. Comput. 221, 169-176 (2013) View ArticleMathSciNetGoogle Scholar
- Guo, S, Qi, F, Srivastava, HM: A class of logarithmically completely monotonic functions related to the gamma function with applications. Integral Transforms Spec. Funct. 23, 557-566 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Guo, S, Qi, F, Srivastava, HM: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function. Appl. Math. Comput. 197, 768-774 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Guo, S, Srivastava, HM: A class of logarithmically completely monotonic functions. Appl. Math. Lett. 21, 1134-1141 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Guo, S, Qi, F, Srivastava, HM: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic. Integral Transforms Spec. Funct. 18, 819-826 (2007) View ArticleMathSciNetGoogle Scholar