Some New Optimal Bounds for Wallis Ratio

Yin Chen *

Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada.

*Author to whom correspondence should be addressed.


Abstract

Wallis ratio can be expressed as an asymptotic expansion using Stirling series and Bernoulli numbers. We prove the general inequalities for Wallis ratio for arbitrary number of terms in the asymptotic expansion. We show that the coefficients in the asymptotic expansion are the best possible.

Keywords: Wallis ratio, bernoulli numbers, stirling series


How to Cite

Chen, Yin. 2023. “Some New Optimal Bounds for Wallis Ratio”. Asian Research Journal of Mathematics 19 (10):169-78. https://doi.org/10.9734/arjom/2023/v19i10739.

Downloads

Download data is not yet available.

References

Benhamou E. Distribution and statistics of the Sharpe Ratio; 2021. ffhal-03207169

Chen CP, Qi F. The best bounds in Wallis inequality. Proc. Amer. Math. Soc. 2005;133(2):397-401.

Guo S, Xu J, Qi F. Some exact constants for the approximation of the quantity in the Wallis’s formula. J. Inequal. Appl. 2013;2013(67):7.

Koumandos S. Remarks on a paper by Chao-Ping Chen and Feng Qi. Proc. Amer. Math. Soc. 2005;134(5):1365-1367.

Mortici C. Completely monotone functions and the Wallis ratio. Applied Mathematics Letters. 2012;25:717- 722.

Zhao Y, Wu Q. Wallis inequality with a parameter. RGMIA Res. Rep. Coll. 2006;7(2):56.

You X, Chen DR. Sharp approximation formulas and inequalities for the Wallis ratio by continued fraction. J. Math. Anal. Appl. 2017;455(2):1743-1748.

Qi F. Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010;2010(493058):84.

Qi F, Luo QM. Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezovic-Giordano-Pecaric’s theorem. J. Inequal. Appl. 2013;2013(542):20.

Qi F, Luo QM. Bounds for the ratio of two gamma functions - From Wendel’s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 2012;6(2):132-158.

Xu A. Asymptotic expansions related to the Wallis ratio based on the Bell polynomials. Asian Research Journal of Mathematics. 2022;18(11):342-350.

Jordan C. Calculus of finite difference. New York: Chelsea; 1965.

Liu G, Luo H. Some identities involving Bernoulli numbers. Fibonacci Quarterly. 2005;43(3):208-212.

Alzer H. Sharp bounds for the Bernoulli Numbers. Arch. Math. 2000;74(3):207-211.

D’Aniello C. On some inequalities for the Bernoulli numbers. Rendiconti del Circolo Matematico di Palermo. 1994;43(3):329-332.

Ge H. New sharp bounds for the Bernoulli Numbers and refinement of Becker-Stark inequalities. Journal of Applied Mathemtaics. 2012;2012(10):7.