# Some Inequalities for Modified Bessel Functions

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

**2010**:253035

**DOI: **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 .

## 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

## References

- Bordelon DJ: Solution to problem 72–15.
*SIAM Review*1973, 15: 666–668.MathSciNetGoogle Scholar - Ross DK: Solution to problem 72–15.
*SIAM Review*1973, 15: 668–670.Google Scholar - Laforgia A: Bounds for modified Bessel functions.
*Journal of Computational and Applied Mathematics*1991, 34(3):263–267. 10.1016/0377-0427(91)90087-ZMATHMathSciNetView ArticleGoogle Scholar - Baricz Á: Bounds for modified Bessel functions of the first and second kind.
*Proceedings of the Edinburgh Mathematical Society*. In press Proceedings of the Edinburgh Mathematical Society. In press - Baricz Á: Tight bounds for the generalized Marcum -function.
*Journal of Mathematical Analysis and Applications*2009, 360(1):265–277. 10.1016/j.jmaa.2009.06.055MATHMathSciNetView ArticleGoogle Scholar - Baricz Á, Sun Y: New bounds for the generalized marcum -function.
*IEEE Transactions on Information Theory*2009, 55(7):3091–3100.MathSciNetView ArticleGoogle Scholar - Marcum JI: A statistical theory of target detection by pulsed radar.
*IRE Transactions on Information Theory*1960, 6: 59–267. 10.1109/TIT.1960.1057560MathSciNetView ArticleGoogle Scholar - Marcum JI, Swerling P: Studies of target detection by pulsed radar.
*IEEE Transactions on Information Theory*1960, 6: 227–228.View ArticleGoogle Scholar - Nuttall AH: Some integrals involving the function.
*IEEE Transactions on Information Theory*1975, 21: 95–96. 10.1109/TIT.1975.1055327MATHMathSciNetView ArticleGoogle Scholar - Simon MK, Alouini MS:
*Digital Communication over Fadding Channels: A Unified Approach to Performance Analysis*. John Wiley & Sons, New York, NY, USA; 2000.View ArticleGoogle Scholar - Abramowitz M, Stegun IA:
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series*.*Volume 55*. U.S. Government Printing Office, Washington, DC, USA; 1964:xiv+1046.Google Scholar - Lorch L: Monotonicity of the zeros of a cross product of Bessel functions.
*Methods and Applications of Analysis*1994, 1(1):75–80.MATHMathSciNetGoogle Scholar - Ismail MEH, Muldoon ME: Monotonicity of the zeros of a cross-product of Bessel functions.
*SIAM Journal on Mathematical Analysis*1978, 9(4):759–767. 10.1137/0509055MATHMathSciNetView ArticleGoogle Scholar - Laforgia A, Natalini P: On some Turán-type inequalities.
*Journal of Inequalities and Applications*2006, 2006:-6.Google Scholar - Baricz Á: On a product of modified Bessel functions.
*Proceedings of the American Mathematical Society*2009, 137(1):189–193.MATHMathSciNetView ArticleGoogle Scholar - Soni RP: On an inequality for modified Bessel functions.
*Journal of Mathematical Physics*1965, 44: 406–407.MATHMathSciNetView ArticleGoogle Scholar - Näsell I: Inequalities for modified Bessel functions.
*Mathematics of Computation*1974, 28: 253–256.MathSciNetView ArticleGoogle Scholar - Jones AL: An extension of an inequality involving modified Bessel functions.
*Journal of Mathematical Physics*1968, 47: 220–221.MATHView ArticleGoogle Scholar - Simpson HC, Spector SJ: Some monotonicity results for ratios of modified Bessel functions.
*Quarterly of Applied Mathematics*1984, 42(1):95–98.MATHMathSciNetGoogle Scholar - Simpson HC, Spector SJ: On barrelling for a special material in finite elasticity.
*Quarterly of Applied Mathematics*1984, 42(1):99–111.MATHMathSciNetGoogle Scholar

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