IMPROVEMENT OF LOWER BOUND IN WALLIS’ INEQUALITY

CHAO-PING CHEN AND FENG QI

Abstract. For all natural number n, we have

\frac{1}{\sqrt{π(n + 2/7)}} < \frac{(2n − 1)!!}{(2n)!!} < \frac{1}{\sqrt{π(n + 1/4)}} .

The constants \frac{2}{7} and \frac{1}{4} are the best possible. From this, the well-known Wallis’ inequality is improved.

1. Introduction

Let

P_{n} = \frac{(2n − 1)!!}{(2n)!!},  (1)

then we have

\frac{1}{2\sqrt{n}} < \sqrt{\frac{2}{π}} · \frac{1}{\sqrt{2n + 1}} < P_{n} < \frac{2}{\sqrt{π}} · \frac{1}{\sqrt{4n + 1}} < \frac{1}{\sqrt{3n + 1}} < \frac{1}{\sqrt{2n + 1}} < \frac{1}{\sqrt{2n}} (2)

for n > 1. The inequality (2) is called Wallis’ inequality in [3, p. 103].

The lower and upper bounds of P_{n} in (2) are always cited and applied by mathematicians. The smallest upper bound \frac{2}{\sqrt{π}} · \frac{1}{\sqrt{4n+1}} and the largest lower bound \sqrt{\frac{2}{π}} · \frac{1}{\sqrt{2n+1}} in (2), that is, the following inequalities

\sqrt{\frac{2}{π}} · \frac{1}{\sqrt{2n + 1}} < P_{n} < \frac{2}{\sqrt{π}} · \frac{1}{\sqrt{4n + 1}}  (3)

are obtained by N. D. Kazarinoff. See [2, pp. 47–48 and pp. 65–67].

Date: July 16, 2002. 1991 Mathematics Subject Classification. Primary 26D15; Secondary 26A48. The authors were supported in part by NNSF (#10001016) of China, SF for the Prominent

Youth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, NSF of Henan Province (#004051800), SF for Pure Research of Natural Science of the Education Department of Henan Province (#1999110004), Doctor Fund of Jiaozuo Institute of Technology, China.

This paper was typeset using AMS-L^{A}TEX.

2 CH.-P. CHEN AND F. QI

We can rewrite inequality (3) as

\frac{1}{\sqrt{π(n + \frac{1}{2})}} < P_{n} < \frac{1}{\sqrt{π(n + \frac{1}{4})}}  (4)

for n ∈ ℕ. See [1, p. 259].

It is well-known that factorials and their ‘continuous’ extension play an eminent role, for instance, in Combinatorics, Graph Theory, and Special Functions.

In this article, we will refine inequality (4). More precisely, we will ask for two best possible constants a and b such that the following double inequality

\frac{1}{\sqrt{π(n + a)}} < P_{n} < \frac{1}{\sqrt{π(n + b)}}  (5)

holds for all natural number n. In other words, the constants a and b can not be replaced by smaller or larger numbers in (5) respectively.

2. Lemma

Lemma 1. Let f(x) = \frac{Γ(x+1)}{Γ(x+ \frac{1}{2} )\sqrt{x+c}}, where c is a constant. Then lim_{x→∞} f(x) = 1.

Proof. Taking logarithm yields

ln f(x) = ln Γ(x + 1) − \frac{1}{2} ln(x + c) − ln Γ(x + \frac{1}{2}).  (6)

The following approximating expansion of Γ(x) is given in [5, p. 107]:

ln Γ(x) = (x − \frac{1}{2}) ln x − x + \frac{1}{2} ln (2π) + ∑_{r=1}^{n} \frac{(−1)^{r−1}B_{r}}{2r(2r − 1)} x^{−2r+1} + O(x^{−2n−1}), (7)

where B_{r} denotes the r-th Bernoulli number which is defined in [5, p. 1] by

t (\frac{1}{e^{t} − 1} + \frac{1}{2}) = 1 + ∑_{n=1}^{∞}(−1)^{n−1} \frac{t^{2n}}{(2n)!}B_{n}.  (8)

Utilizing (7) in (6) and simplifying can give the following

ln f(x) = x ln \frac{x + 1}{x + \frac{1}{2}} + \frac{1}{2} ln \frac{x + 1}{x + c} − \frac{1}{2} + O (\frac{1}{x}) → 0 (x → ∞),  (9)

and then we have lim_{x→∞} f(x) = 1. The proof is complete.

Corollary 1. For any constant c, we have

lim_{n→∞} \frac{Γ(n + 1)}{Γ(n + \frac{1}{2})\sqrt{n + c}} = 1.  (10)

Proof. This is a special case of Lemma 1.

IMPROVEMENT OF LOWER BOUND IN WALLIS’ INEQUALITY 3

3. Main results

Now we will give the main results of this paper. Theorem 1. For all natural number n, we have

\frac{1}{\sqrt{π(n + \frac{2}{7})}} < \frac{(2n − 1)!!}{(2n)!!} < \frac{1}{\sqrt{π(n + \frac{1}{4})}}.  (11)

The constants \frac{2}{7} and \frac{1}{4} are the best possible, which means that they cannot be replaced by smaller number and larger number, respectively.

Proof. Let

Ω_{n} = \frac{Γ(n + 1)}{Γ(n + \frac{1}{2})\sqrt{n + c}}  (12)

for n ∈ ℕ. Since

Γ(n + 1) = n!, Γ(n + \frac{1}{2}) = \frac{(2n − 1)!!}{2^{n}} \sqrt{π}, 2^{n}n! = (2n)!!,

then we have

Ω_{n} = \frac{1}{\sqrt{π(n + c)}} · \frac{1}{P_{n}} ,  (13)

furthermore,

\frac{Ω_{n+1}}{Ω_{n}} = \frac{2(n + 1)}{2n + 1}\sqrt{\frac{n + c}{n + 1 + c}}.  (14)

It is easy to see that the sequence {Ω_{n}}_{n=1}^{∞} decreases strictly if and only if

\frac{2(n + 1)}{2n + 1}\sqrt{\frac{n + c}{n + 1 + c}} < 1

holds for all natural number n ∈ ℕ, which is equivalent to −1 < c ≤ \frac{1}{4}. Since lim_{n→∞} Ω_{n} = 1 from (10), thus we have

Ω_{n} = \frac{1}{\sqrt{π(n + c)}} · \frac{1}{P_{n}} > 1,

which is equivalent to

P_{n} < \frac{1}{\sqrt{π(n + c)}}  (15)

for n ∈ ℕ and −1 < c ≤ \frac{1}{4}.

From (14), it follows that the sequence {Ω_{n}}_{n=1}^{∞} increases strictly if and only if

\frac{2(n + 1)}{2n + 1}\sqrt{\frac{n + c}{n + 1 + c}} > 1  (16)

holds for all natural number n ∈ ℕ, which is equivalent to c > \frac{2}{7}. Therefore, if c > \frac{2}{7}, the following inequality

P_{n} > \frac{1}{\sqrt{π(n + c)}}  (17)

holds for all n ∈ ℕ.

For c = \frac{2}{7}, inequality (16) holds for n ≥ 2. This means that the sequence

\{\frac{1}{P_{n}\sqrt{π(n + \frac{2}{7})}}\}_{n=2}^{∞}

is strictly increasing and tends to 1 as n → ∞. Therefore

P_{n} > \frac{1}{\sqrt{π(n + \frac{2}{7})}}  (18)

holds for all n ≥ 2. Direct computing reveals that inequality (18) is valid for n = 1. Hence, inequality (17) holds for all natural number n and c ≥ \frac{2}{7}.

In conclusion, the following double inequality

\frac{1}{\sqrt{π(n + a)}} < P_{n} < \frac{1}{\sqrt{π(n + b)}}  (19)

holds for all n ∈ ℕ and for all a ≥ \frac{2}{7} and b ≤ \frac{1}{4}. The proof is complete.

References

[1] P. S. Bullen, A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics 97, Addison Wesley Longman Limited, 1998.

[2] N. D. Kazarrinoff, Analytic Inequalities, Holt, Rhinehart and Winston, New York, 1961.

[3] J.-Ch. Kuang, Chángyòng Bùděngshì (Applied Inequalities), 2nd edition, Hunan Education Press, Changsha, China, 1993. (Chinese)

[4] F. Qi, L.-H. Cui, and S.-L. Xu, Some inequalities constructed by Tchebysheff’s integral inequality, Math. Inequal. Appl. 2 (1999), no. 4, 517–528.

[5] Zh.-X. Wang and D.-R. Guo, Tèshū Hánshù Gàilùn (Introduction to Special Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000. (Chinese)

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, #142, Mid-Jiefang Road, Jiaozuo City, Henan 454000, CHINA

   E-mail address: qifeng@jzit.edu.cn
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