- Research Article
- Open Access
Some Inequalities for Modified Bessel Functions
© A. Laforgia and P. Natalini. 2010
- Received: 15 October 2009
- Accepted: 28 December 2009
- Published: 24 January 2010
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 .
- Generalize Function
- Lower Bound
- Numerical Experiment
- Performance Analysis
- Bessel Function
have been established; see also 
(see also ). 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
The results are given by the following theorems.
In particular, for , the inequality holds also true.
In particular, for , the inequality holds also true.
Proof of Theorem 1.1.
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.
For , Jones  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).
By means the integral formula [11, page 181]
Since when , only in this case the above upper bound for improves the (1.13) one.
in particular, for , we also have .
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).
Let be . Putting and , one has , for all .
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.
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 .
For a survey on inequalities of the type (3.2) and (3.3) see .
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 ).
This work was sponsored by Ministero dell'Universitá e della Ricerca Scientifica Grant no. 2006090295.
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