Some Inequalities for Modified Bessel Functions
 Andrea Laforgia^{1} and
 Pierpaolo Natalini^{1}Email author
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 Turántype 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 Turántype 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.
In particular, for , the inequality holds also true.
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 Turántype 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 get , for .
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.
in particular, for , we also have .
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 .
Table 1


 









Remark.
Example.
Example.
Table 2


 





Table 3


 





Example.
Table 4


 





Remark.
For a survey on inequalities of the type (3.2) and (3.3) see [4].
Table 5


 













Table 6



 



 —  — 










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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