- Research
- Open Access
- Published:

# A sharp two-sided inequality for bounding the Wallis ratio

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 43 (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.

## Introduction and main results

For \(n\in\mathbb{N}\) (the set of all positive integers), the double factorial \(n!!\) is defined by

where in (1) the floor function \(\lfloor t\rfloor\) denotes the largest integer less than or equal to *t*. For our own convenience, in what follows, we denote the ratio of two neighboring double factorials by

which is called the Wallis ratio in the literature.

The Wallis ratio \(W_{n}\) can be represented as follows (see [1, p.258]):

where in (3) \(\Gamma(x)\) is the classical Euler’s gamma function defined for \(x>0\) by

In [2] the author proved, for all \(n\in\mathbb{N}\),

In [3], the following result was established:

For \(n\in\mathbb{N}\) and \(\varepsilon\in(0,\frac{1}{2})\),

The right-hand inequality in (6) holds for \(n > n^{*}\), where \(n^{*}\) is the maximal root on

We also note that in [4] the authors proved the result below.

For all \(n\in\mathbb{N}\)

which is equivalent to the following:

In this article we shall establish a sharp two-sided inequality for bounding the Wallis ratio in the form

where in (9) the constants \(C_{1}>0\) and \(C_{2}>0\) are best possible. This means that the constant \(C_{1}\) in (9) cannot be replaced by a number which is greater than \(C_{1}\) and the constant \(C_{2}\) in (9) cannot be replaced by a number which is less than \(C_{2}\). An asymptotic formula for the Wallis ratio is also given.

Our main result may be stated as the following theorem.

### Theorem 1

*For all*
\(n\ge2\),

*The constants*
\((\frac{2}{3} )^{3/2}\)
*and*
\(\sqrt{\frac{e}{\pi}} \)
*in* (10) *are best possible*. *Furthermore*, *the asymptotic formula*

*is valid*.

## Proof of main result

We are now in a position to prove our main result stated in Theorem 1.

### Proof of Theorem 1

Define

Taking the logarithm of \(f(x)\) and then differentiating yield

In (14)

It is well known that (see [5, p.892])

Hence,

which means that \(\ln f(x)\), and thus \(f(x)\), is strictly increasing on \((0,\infty)\).

It is easy to see that

Since (see [6, p.20])

which implies

Since the function \(f(x)\) is strictly increasing from \((0,\infty)\) onto \((0,\frac{1}{\sqrt{2\pi} } )\) and

we obtain

and

The lower and upper bounds in (21) are best possible.

Also we define

Since the function \(h(x)\) is strictly increasing from \((1,\infty)\) onto \((0,\sqrt{2\pi} )\) (see [7, Theorem 1.3]) and in view of (20), we obtain

and

It is well known that [1, p.258] for all \(n\in\mathbb{N}\),

By using (26), after some algebra, (24) and (25) can be rewritten, respectively, as

and

The constants \(\frac{e^{2}}{3\sqrt{3} }\) and \(\sqrt{2e} \) in (27) are best possible.

Combining (21) and (27) yields

The constants \((\frac{2}{3} )^{3/2}\) and \(\sqrt{\frac{e}{\pi}} \) in (29) are best possible. From (29) the inequality (10) follows.

which is equivalent to the asymptotic formula (11). The proof of Theorem 1 is thus completed. □

### Remark 1

Some related functions associated with \(f(x)\), defined by (12), were proved [8–12] to be logarithmically completely monotonic.

## References

- 1.
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)

- 2.
Mortici, C: Completely monotone functions and the Wallis ratio. Appl. Math. Lett.

**25**, 717-722 (2012) - 3.
Zhao, Y, Wu, Q: Wallis inequality with a parameter. J. Inequal. Pure Appl. Math.

**7**, 56 (2006) - 4.
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) - 5.
Gradshteyn, IS, Ryzhik, IM: Table of Integrals, Series, and Products, 6th edn. Academic Press, New York (2000)

- 6.
Andrews, GE, Askey, R, Roy, R: Special Functions. Cambridge University Press, Cambridge (1999)

- 7.
Guo, S: Monotonicity and concavity properties of some functions involving the gamma function with applications. J. Inequal. Pure Appl. Math.

**7**, 45 (2006) - 8.
Guo, S: Logarithmically completely monotonic functions and applications. Appl. Math. Comput.

**221**, 169-176 (2013) - 9.
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) - 10.
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) - 11.
Guo, S, Srivastava, HM: A class of logarithmically completely monotonic functions. Appl. Math. Lett.

**21**, 1134-1141 (2008) - 12.
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)

## 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.

## Author information

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All the authors contributed to the writing of the present article. They also read and approved the final manuscript.

## Rights and permissions

**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.

## About this article

#### Received

#### Accepted

#### Published

#### DOI

### MSC

- 11B65
- 41A44
- 05A10
- 26D20
- 33B15
- 41A60

### Keywords

- estimation
- inequality
- asymptotic formula
- Wallis ratio
- gamma function