Research

# Sharp Cusa and Becker-Stark inequalities

Chao-Ping Chen1* and Wing-Sum Cheung2

Author Affiliations

1 School of Mathematics and Informatics, Henan Polytechnic, University, Jiaozuo City 454003, Henan Province, People's Republic of China

2 Department of Mathematics, the University of Hong Kong, Pokfulam Road, Hong Kong, China

For all author emails, please log on.

Journal of Inequalities and Applications 2011, 2011:136 doi:10.1186/1029-242X-2011-136

 Received: 8 June 2011 Accepted: 7 December 2011 Published: 7 December 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We determine the best possible constants θ,ϑ,α and β such that the inequalities

and

are valid for 0 < × < π/2. Our results sharpen inequalities presented by Cusa, Becker and Stark.

Mathematics Subject Classification (2000): 26D05.

##### Keywords:
Inequalities; trigonometric functions

### 1. Introduction

For 0 < × < π/2, it is known in the literature that

(1)

Inequality (1) was first mentioned by the German philosopher and theologian Nicolaus de Cusa (1401-1464), by a geometrical method. A rigorous proof of inequality (1) was given by Huygens [1], who used (1) to estimate the number π. The inequality is now known as Cusa's inequality [2-5]. Further interesting historical facts about the inequality (1) can be found in [2].

It is the first aim of present paper to establish sharp Cusa's inequality.

Theorem 1. For 0 < × < π/2,

(2)

with the best possible constants

Becker and Stark [6] obtained the inequalities

(3)

The constant 8 and π2 are the best possible.

Zhu and Hua [7] established a general refinement of the Becker-Stark inequalities by using the power series expansion of the tangent function via Bernoulli numbers and the property of a function involving Riemann's zeta one. Zhu [8] extended the tangent function to Bessel functions.

It is the second aim of present paper to establish sharp Becker-Stark inequality.

Theorem 2. For 0 < × < π/2,

(4)

with the best possible constants

Remark 1. There is no strict comparison between the two lower bounds and

in (3) and (4).

The following lemma is needed in our present investigation.

Lemma 1 ([9-11]). Let - < a < b < ∞, and f, g : [a, b] → ℝ be continuous on [a, b] and differentiable in (a, b). Suppose g' ≠ 0 on (a; b). If f'(x)/g' (x) is increasing (decreasing) on (a, b), then so are

If f'(x) = g'(x) is strictly monotone, then the monotonicity in the conclusion is also strict.

### 2. Proofs of Theorems 1 and 2

Proof of Theorem [1]. Consider the function f(x) defined by

For 0 < x < π/2, let

Then,

where

Differentiating with respect to x yields

Elementary calculations reveal that

where

By using the power series expansions of sine and cosine functions, we find that

where

Elementary calculations reveal that, for 0 < × < π/2 and n ≥ 4,

Hence, for fixed x ∈ (0, π/2), the sequence nun(x) is strictly decreasing with regard to n ≥ 4. Hence, for 0 < × < π/2,

and therefore, the functions F5(x) and are both strictly increasing on (0, π/2).

By Lemma 1, the function

is strictly increasing on (0, π/2). By Lemma 1, the function

is strictly increasing on (0, π/2), and we have

By rearranging terms in the last expression, Theorem 1 follows.

Proof of Theorem 2. Consider the function f(x) defined by

For 0 < x < π/2, let

Then,

Elementary calculations reveal that

Motivated by the investigations in [12], we are in a position to prove h(x) > 0 for x ∈ (0, π/2).Let

Where λ and μ are constants determined with limits:

Using Maple, we determine Taylor approximation for the function H(x) by the polynomial of the first order:

which has a bound of absolute error

for values x ∈ [0,π/2]. It is true that

for x ∈ [0, π/2]. Hence, for x ∈ [0, π/2], it is true that H (x) > 0 and, therefore, h (x) > 0 and g'(x) > 0 for x ∈ [0, π/2]. Therefore, the function is strictly increasing on. (0, π/2).By Lemma 1, the function

is strictly increasing on (0, π/2), and we have

By rearranging terms in the last expression, Theorem 2 follows.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript

### Acknowledgements

Research is supported in part by the Research Grants Council of the Hong Kong SAR, Project No. HKU7016/07P.

### References

1. Huygens, C: Oeuvres Completes 1888-1940. In: Société Hollondaise des Science, Haga PubMed Abstract | Publisher Full Text | PubMed Central Full Text

2. Sandor, J, Bencze, M: On Huygens' trigonometric inequality. RGMIA Res Rep Collect. 8(3), Article 14 (2005)

3. Zhu, L: A source of inequalities for circular functions. Comput Math Appl. 58, 1998–2004 (2009). Publisher Full Text

4. Neuman, E, Sándor, J: On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities. Math Inequal Appl. 13, 715–723 (2010)

5. Mortiti, C: The natural approach of Wilker-Cusa-Huygens inequalities. Math Inequal Appl. 14, 535–541 (2011)

6. Becker, M, Strak, EL: On a hierarchy of quolynomial inequalities for tanx. Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz.(602-633), 133–138 (1978)

7. Zhu, L, Hua, JK: Sharpening the Becker-Stark inequalities. J Inequal Appl Article ID 931275 (2010)

8. Zhu, L: Sharp Becker-Stark-type inequalities for Bessel functions. J Inequal Appl Article ID 838740 (2010)

9. Anderson, GD, Qiu, SL, Vamanamurthy, MK, Vuorinen, M: Generalized elliptic integral and modular equations. Pac J Math. 192, 1–37 (2000). Publisher Full Text

10. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Conformal Invariants, Inequalities, and Quasiconformal Maps, New York (1997)

11. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Monotonicity of Some Functions in Calculus. Available online at www.math.auckland.ac.nz/Research/Reports/Series/538.pdf

12. Malešević, BJ: One method for proving inequalities by computer. J Ineq Appl Article ID 78691 (2007)