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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

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

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2011/1/136


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

© 2011 Chen and Cheung; licensee Springer.

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M1">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M2">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M3">View MathML</a>

(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,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M4">View MathML</a>

(2)

with the best possible constants

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M5">View MathML</a>

Becker and Stark [6] obtained the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M6">View MathML</a>

(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,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M7">View MathML</a>

(4)

with the best possible constants

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M8">View MathML</a>

Remark 1. There is no strict comparison between the two lower bounds<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M9">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M10">View MathML</a>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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M11">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M12">View MathML</a>

For 0 < x < π/2, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M13">View MathML</a>

Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M14">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M15">View MathML</a>

Differentiating with respect to x yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M16">View MathML</a>

Elementary calculations reveal that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M17">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M18">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M19">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M20">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M21">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M22">View MathML</a>

and therefore, the functions F5(x) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M23">View MathML</a>are both strictly increasing on (0, π/2).

By Lemma 1, the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M24">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M25">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M26">View MathML</a>

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

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M27">View MathML</a>

For 0 < x < π/2, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M28">View MathML</a>

Then,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M29">View MathML</a>

Elementary calculations reveal that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M30">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M31">View MathML</a>

Where λ and μ are constants determined with limits:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M32">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M33">View MathML</a>

which has a bound of absolute error

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M34">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M35">View MathML</a>

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M36">View MathML</a>is strictly increasing on. (0, π/2).By Lemma 1, the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M37">View MathML</a>

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2011/1/136/mathml/M38">View MathML</a>

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.

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