Sharp Becker-Stark-Type Inequalities for Bessel Functions

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.