# Some Inequalities for Modified Bessel Functions

- Andrea Laforgia
^{1}and - Pierpaolo Natalini
^{1}Email author

**2010**:253035

https://doi.org/10.1155/2010/253035

© A. Laforgia and P. Natalini. 2010

**Received: **15 October 2009

**Accepted: **28 December 2009

**Published: **24 January 2010

## Abstract

We denote by and the Bessel functions of the first and third kinds, respectively. Motivated by the relevance of the function , , in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector (1984), we establish new inequalities for . The results are based on the recurrence relations for and and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for .

## Keywords

## 1. Introduction

Inequalities for modified Bessel functions and have been established by many authors. For example, Bordelon [1] and Ross [2] proved the bounds

The lower bound was also proved by Laforgia [3] for larger domain . In [3] also the following bounds:

have been established; see also [4]

(see also [6]). This generalized function and the classical one, , are widely used in the electronic field, in particular in radar communications [7, 8] and in error performance analysis of multichannel dealing with partially coherent, differentially coherent, and noncoherent detections over fading channels [7, 9, 10].

The results obtained in this paper are proved as consequence of the recurrence relations [11, page 376; 9.6.26]

and the Turán-type inequalities

proved in [12, 13], respectively (see also [14] for (1.9)). Inequalities (1.8)-(1.9) have been used, recently, by Baricz in [15], to prove, in different way, the known inequalities

The results are given by the following theorems.

Theorem 1.1.

In particular, for , the inequality holds also true.

Theorem.

## 2. The Proofs

Proof of Theorem 1.1.

proved by Soni for [16], and extended by Näsell to [17].

To prove the lower bound in (1.12), we substitute the function given by (1.6) in the Turán-type inequality (1.8). We get, for ,

which is the desired result.

Remark.

For , Jones [18] proved stronger result than (2.1) that the function decreases with respect to , when .

Proof of Theorem 1.2.

We substitute the function given by (1.7) in (1.9). We get

which is the desired result (1.13).

Remark.

By means the integral formula [11, page 181]

Since when , only in this case the above upper bound for improves the (1.13) one.

Remark.

## 3. Numerical Considerations

Baricz obtained, for each , the following similar lower bound for the ratio (see [5, formula (5)])

where is the unique simple positive root of the equation . Inequality (3.1) is reversed when . It is possible to prove that, for , our lower bound in (1.12) for the ratio provides an improvement of (3.1).

Proposition.

Let be . Putting and , one has , for all .

Proof.

From the inequality we obtain, by simple calculations, the following one which is satisfied for all when .

We report here some numerical experiments, computed by using mathematica.

Example.

In the first case we assume . In Figure 1 we report the graphics of the functions (solid line) and the respective lower bounds (short dashed line) and (long dashed line) on the interval .

Remark.

Example.

Example.

Example.

Remark.

For a survey on inequalities of the type (3.2) and (3.3) see [4].

By the values reported on Table 5 it seems that is a lower bound much more stringent with respect to for every (moreover we recall that (3.2) holds true also for ), while by the values reported on Table 6 it seems that is a lower bound more stringent with respect to for (but we recall that (3.3) holds true also for and ).

## Declarations

### Acknowledgment

This work was sponsored by Ministero dell'Universitá e della Ricerca Scientifica Grant no. 2006090295.

## Authors’ Affiliations

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