Open Access

Sharp Becker-Stark-Type Inequalities for Bessel Functions

Journal of Inequalities and Applications20102010:838740

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

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.

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
(1.1)

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

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

Theorem 1.2.

Let ; then
(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.

Let , and be a natural number. Then
(1.3)
holds, where , and
(1.4)

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
(1.5)
where, is the well- known Pochhammer (or Appell) symbol, and defined by [4, page 40]
(1.6)

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
(2.1)

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
(2.2)
(2.3)

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
(2.4)

where .

Proof.

By Lemma 2.1 and (2.3) in Lemma 2.2, we have
(2.5)

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
(3.1)
holds, where and
(3.2)

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

Proof of Theorem 3.1.

Let
(3.3)
Then by Lemma 2.3, we have
(3.4)

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

(1)
Department of Mathematics, Zhejiang Gongshang University

References

  1. 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
  2. Kuang JC: Applied Inequalities. 3rd edition. Shandong Science and Technology Press, Jinan, China; 2004.Google Scholar
  3. Zhu L, Hua JK: “Sharpening the Becker-Stark inequalities”, Journal of Inequalities and Applications 2010, 2010:-4.Google Scholar
  4. Watson GN: A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, UK; 1995:viii+804.Google Scholar
  5. Kishore N: “The Rayleigh function”, Proceedings of the American Mathematical Society 1963, 14: 527–533. 10.1090/S0002-9939-1963-0151649-2MathSciNetView ArticleMATHGoogle Scholar
  6. Baricz Á, Wu S: “Sharp exponential Redheffer-type inequalities for Bessel functions”, Publicationes Mathematicae Debrecen 2009, 74(3–4):257–278.MathSciNetMATHGoogle Scholar

Copyright

© Ling Zhu. 2010

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.