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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The best bounds in Wallis' inequality


Authors: Chao-Ping Chen and Feng Qi
Journal: Proc. Amer. Math. Soc. 133 (2005), 397-401
MSC (2000): Primary 05A10, 26D20; Secondary 33B15
Published electronically: August 30, 2004
MathSciNet review: 2093060
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Abstract | References | Similar Articles | Additional Information

Abstract: For all natural numbers $n$, let $n!!$ denote a double factorial. Then

\begin{displaymath}\frac1{\sqrt{\pi\bigl(n+\frac4{\pi}-1\bigr)}}\leq \frac{(2n-1)!!}{(2n)!!}<\frac1{\sqrt{\pi\bigl(n+\frac14\bigr)}}. \end{displaymath}

The constants $\frac{4}{\pi}-1$ and $\frac14$ are the best possible. From this, the well-known Wallis' inequality is improved.


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Additional Information

Chao-Ping Chen
Affiliation: Department of Applied Mathematics and Informatics, Research Institute of Applied Mathematics, Henan Polytechnic University, Jiaozuo City, Henan 454000, People’s Republic of China
Email: chenchaoping@hpu.edu.cn

Feng Qi
Affiliation: Department of Applied Mathematics and Informatics, Research Institute of Applied Mathematics, Henan Polytechnic University, Jiaozuo City, Henan 454000, People’s Republic of China
Email: qifeng@hpu.edu.cn, fengqi618@member.ams.org

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07499-4
PII: S 0002-9939(04)07499-4
Keywords: Wallis' inequality, best bound, gamma function, monotonicity
Received by editor(s): August 3, 2002
Received by editor(s) in revised form: June 23, 2003, and September 27, 2003
Published electronically: August 30, 2004
Additional Notes: The authors were supported in part by NSF (#10001016) of China, SF for the Prominent Youth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), Doctor Fund of Jiaozuo Institute of Technology, China
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society