On approximating the modified Bessel function of the second kind.

In the article, we prove that the double inequalities [Formula: see text] hold for all [Formula: see text] if and only if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the modified Bessel function of the second kind. As applications, we provide bounds for [Formula: see text] with [Formula: see text] and present the necessary and sufficient condition such that the function [Formula: see text] is strictly increasing (decreasing) on [Formula: see text].

Introduction

The modified Bessel function of the first kind is a particular solution of the second-order differential equation and it can be expressed by the infinite series

While the modified Bessel function of the second kind is defined by where the right-hand side of the identity of (1.1) is the limiting value in case ν is an integer.

The following integral representation formula and asymptotic formulas for the modified Bessel function of the second kind can be found in the literature [1], 9.6.24, 9.6.8, 9.6.9, 9.7.2:

From (1.2) we clearly see that

Recently, the bounds for the modified Bessel function of the second kind have attracted the attention of many researchers. Luke [2] proved that the double inequality holds for all .

Gaunt [3] proved that the double inequality takes place for all , where is the classical gamma function.

In [4], Segura proved that the double inequality holds for all and .

Bordelon and Ross [5] and Paris [6] provided the inequality for all and .

Laforgia [7] established the inequality for all if , and inequality (1.12) is reversed if .

Baricz [8] presented the inequality for all and .

Motivated by inequality (1.9), in the article, we prove that the double inequality holds for all if and only if and if . As applications, we provide bounds for with and present the necessary and sufficient condition such that the function is strictly increasing (decreasing) on .

Lemmas

In order to prove our main results, we need two lemmas which we present in this section.

Lemma 2.1

See [9]

Let , be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are the functions If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2

The function is strictly increasing from onto .

Proof

Let . Then it follows from (1.6), (1.7) and (2.1) that for all .

Note that (1.3)-(1.5) and (2.1) lead to

Therefore, Lemma 2.2 follows easily from (2.2)-(2.4). □

Main results

Theorem 3.1

Let . Then the double inequality holds for all if and only if and .

Let , be defined by Lemma 2.2, and , and be respectively defined by and

Then from (1.5), (1.7) and (3.1) we clearly see that

It follows from (3.2)-(3.4), Lemmas 2.1 and 2.2 together with L’Hôpital’s rule that the function is strictly increasing on and

Note that (1.3) and (3.2) lead to

Therefore, Theorem 3.1 follows easily from (3.2), (3.5), (3.6) and the monotonicity of . □

Remark 3.2

From Lemma 2.2 we clearly see that the double inequality holds for all if and only if and .

From (1.7) and Remark 3.2 we get Corollary 3.3 immediately.

Corollary 3.3

Let . Then the double inequalities and hold for all if and only if and .

Remark 3.4

Let . Then from inequality (3.7) we know that the function is strictly increasing (decreasing) on if and only if (). We clearly see that the bounds for given in Corollary 3.3 are better than the bounds given in (1.10) for .

From (1.3), (1.5) and Remark 3.4 we get Corollary 3.5 immediately.

Corollary 3.5

The double inequality holds for all if , and inequality (3.8) is reversed if .

Remark 3.4 also leads to Corollary 3.6.

Corollary 3.6

Let . Then the double inequality holds for all if and only if and .

Remark 3.7

We clearly see that the lower bound for in Corollary 3.6 is better than the bounds given in (1.11) and (1.12) for .

Remark 3.8

From the inequality given in [10], (2.20), and the fact that for all we clearly see that the lower bound given in Theorem 3.1 for is better than that given in (1.8) and (1.9). But the upper bound given in Theorem 3.1 is weaker than that given in (1.8).

Remark 3.9

From Theorem 3.1 and Corollary 3.3 we clearly see that there exist and such that for all .

Theorem 3.10

Let , , , , , and and be defined by Then the double inequality holds for all and .

We use mathematical induction to prove inequality (3.10). From Corollary 3.3 we clearly see that inequality (3.10) holds for all and .

Suppose that inequality (3.10) holds for (), that is, Then it follows from (3.9) and (3.11) together with the formula given in [11] that

Inequality (3.12) implies that inequality (3.10) holds for , and the proof of Theorem 3.10 is completed. □

Remark 3.11

Let . Then Theorem 3.10 leads to for all .

Remark 3.12

It is not difficult to verify that for . Therefore, the bounds given in Remark 3.11 are better than the bounds given in (1.10) for .