- Research Article
- Open Access

# Sharp Becker-Stark-Type Inequalities for Bessel Functions

- Ling Zhu
^{1}Email author

**2010**:838740

https://doi.org/10.1155/2010/838740

© Ling Zhu. 2010

**Received:**22 January 2010**Accepted:**23 March 2010**Published:**6 April 2010

## Abstract

We extend the Becker-Stark-type inequalities to the ratio of two normalized Bessel functions of the first kind by using Kishore formula and Rayleigh inequality.

## Keywords

- Natural Number
- Bessel Function
- Basic Fact
- Tangent Function
- Positive Zero

## 1. Introduction

In 1978, Becker and Stark [1] (or see Kuang [2, page 248]) obtained the following two-sided rational approximation for .

Theorem 1.1.

Furthermore, and are the best constants in (1.1).

In recent paper [3], we obtained the following further result.

Theorem 1.2.

Furthermore, and are the best constants in (1.2).

Moreover, the following refinement of the Becker-Stark inequality was established in [3].

Theorem 1.3.

where are the even-indexed Bernoulli numbers. Furthermore, and are the best constants in (1.3).

Particularly for and , respectively, the function reduces to some elementary functions, like [4, page 54] and . In view of that , in Section 3 we shall extend the result of Theorem 1.3 to the ratio of two normalized Bessel functions of the first kind and .

## 2. Some Lemmas

In order to prove our main result in next section, each of the following lemmas will be needed.

Lemma 2.1 (Kishore Formula, see [5, 6]).

where , and is the Rayleigh function of order , which showed in [4, page 502].

Lemma 2.2 (Rayleigh Inequality [5, 6]).

hold.

Lemma 2.3.

where .

Proof.

where , which follows from (2.2) in Lemma 2.2.

## 3. Main Result and Its Proof

Theorem 3.1.

Furthermore, and are the best constants in (3.1).

Proof of Theorem 3.1.

Since for by Lemma 2.3, is decreasing on .

At the same time, in view of that we have that by (3.3), and by (3.4), so and are the best constants in (3.1).

Remark 3.2.

Let in Theorem 3.1; we obtain Theorem 1.3.

## Authors’ Affiliations

## References

- Becker M, Stark EL: “On a hierarchy of quolynomial inequalities for tanx”,
*University of Beograd Publikacije Elektrotehnicki Fakultet. Serija Matematika i fizika*1978, (602–633):133–138.MathSciNetMATHGoogle Scholar - Kuang JC:
*Applied Inequalities*. 3rd edition. Shandong Science and Technology Press, Jinan, China; 2004.Google Scholar - Zhu L, Hua JK: “Sharpening the Becker-Stark inequalities”,
*Journal of Inequalities and Applications*2010, 2010:-4.Google Scholar - Watson GN:
*A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library*. Cambridge University Press, Cambridge, UK; 1995:viii+804.Google Scholar - Kishore N: “The Rayleigh function”,
*Proceedings of the American Mathematical Society*1963, 14: 527–533. 10.1090/S0002-9939-1963-0151649-2MathSciNetView ArticleMATHGoogle Scholar - Baricz Á, Wu S: “Sharp exponential Redheffer-type inequalities for Bessel functions”,
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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.