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# Sharp Becker-Stark-Type Inequalities for Bessel Functions

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 838740 (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.

## 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.

Let ; then

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

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

Theorem 1.2.

Let ; then

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.

Let , and be a natural number. Then

holds, where , and

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

Our aim of this paper is to extend the tangent function to Bessel functions. To achieve our goal, let us recall some basic facts about Bessel functions. Suppose that and consider the normalized Bessel function of the first kind , defined by

where, is the well- known Pochhammer (or Appell) symbol, and defined by [4, page 40]

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]).

Let , be the th positive zero of the Bessel function of the first kind of order , and . Then

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

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

Let , and be the th positive zero of the Bessel function of the first kind of order , , and is the Rayleigh function of order . Then

hold.

Lemma 2.3.

Let , be the normalized Bessel function of the first kind of order , the th positive zero of the Bessel function of the first kind of order , , the Rayleigh function of order , and . Then

where .

Proof.

By Lemma 2.1 and (2.3) in Lemma 2.2, we have

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

## 3. Main Result and Its Proof

Theorem 3.1.

Let , be the normalized Bessel function of the first kind of order , the th positive zero of the Bessel function of the first kind of order , , the Rayleigh function of order , a natural number, and . Let , and . Then

holds, where and

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

Proof of Theorem 3.1.

Let

Then by Lemma 2.3, we have

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.

## 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.Kuang JC:

*Applied Inequalities*. 3rd edition. Shandong Science and Technology Press, Jinan, China; 2004.Zhu L, Hua JK: “Sharpening the Becker-Stark inequalities”,

*Journal of Inequalities and Applications*2010, 2010:-4.Watson GN:

*A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library*. Cambridge University Press, Cambridge, UK; 1995:viii+804.Kishore N: “The Rayleigh function”,

*Proceedings of the American Mathematical Society*1963, 14: 527–533. 10.1090/S0002-9939-1963-0151649-2Baricz Á, Wu S: “Sharp exponential Redheffer-type inequalities for Bessel functions”,

*Publicationes Mathematicae Debrecen*2009, 74(3–4):257–278.

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

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### Cite this article

Zhu, L. Sharp Becker-Stark-Type Inequalities for Bessel Functions.
*J Inequal Appl* **2010**, 838740 (2010). https://doi.org/10.1155/2010/838740

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

### Keywords

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