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Some Inequalities for Modified Bessel Functions
Journal of Inequalities and Applications volume 2010, Article number: 253035 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ1_HTML.gif)
The lower bound was also proved by Laforgia [3] for larger domain . In [3] also the following bounds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ2_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ4_HTML.gif)
have been established; see also [4]
In this paper we continue our investigations on new inequalities for and
, but now our results refer not only to a function
or
at two different points
and
, as in (1.1)–(1.4), but to two functions
and
(
and
) and, more precisely, to the ratio
. This kind of ratios appears often in applied sciences. Recently, for example, the ratio
has been used by Baricz to prove an important lemma (see [5, Lemma 1]) which provides new lower and upper bounds for the generalized Marcum
-function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ5_HTML.gif)
(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]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ7_HTML.gif)
and the Turán-type inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ11_HTML.gif)
The results are given by the following theorems.
Theorem 1.1.
For real let
be the modified Bessel function of the first kind and order
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ12_HTML.gif)
In particular, for , the inequality
holds also true.
Theorem.
For real let
be the modified Bessel function of the third kind and order
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ13_HTML.gif)
In particular, for , the inequality
holds also true.
2. The Proofs
Proof of Theorem 1.1.
The upper bound for the ratio follows from the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ14_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ15_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ16_HTML.gif)
We denote by
and observe that for
, by (2.1),
. With this notation (2.3) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ17_HTML.gif)
which gives, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ18_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ19_HTML.gif)
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.
The proof is similar to the one used to prove Theorem 1.1. By
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ20_HTML.gif)
we get , for
.
We substitute the function given by (1.7) in (1.9). We get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ21_HTML.gif)
or, equivalently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ22_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ23_HTML.gif)
Finally, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ24_HTML.gif)
which is the desired result (1.13).
Remark.
By means the integral formula [11, page 181]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ25_HTML.gif)
follows immediately the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ26_HTML.gif)
and consequently
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ27_HTML.gif)
Since when
, only in this case the above upper bound for
improves the (1.13) one.
Remark.
We observe that by Theorem 1.1 we obtain an upper bound for the function ,
. The investigations of the properties of
are motivated by some problems of finite elasticity [19, 20]. By (1.12) we find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ28_HTML.gif)
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)])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ29_HTML.gif)
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
.
In Table 1 we report also the respective numerical values of the differences and
in some points
.
![figure 1](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig1_HTML.jpg)
Figure 1
Remark.
By some numerical experiments we can conjecture that the lower bound (3.1) holds true also when and, in particular, for these values of
we have
. See, for example, in Figure 2 the graphics of the functions
(solid line) and the respective lower bounds
(short dashed line) and
(long dashed line) on the interval
when
.
![figure 2](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig2_HTML.jpg)
Figure 2
Example.
In this case we assume , then we report, in Figure 3, the graphics of the functions
(solid line) and the respective lower bounds
(short dashed line) on the interval
.In Table 3 we report also the respective numerical values of the differences
in some points
.
![figure 3](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig3_HTML.jpg)
Figure 3
Example.
Also in this case we assume . In Figure 4 we report the graphics of the functions
(solid line) and the respective upper bound
(short dashed line) on the interval
.In Table 2, we report also the respective numerical values of the difference
in some points
.
![figure 4](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig4_HTML.jpg)
Figure 4
Example.
In this last case we assume . In Figure 5 we report the graphics of the functions
(solid line) and the respective upper bound
(short dashed line) on the interval
. In Table 4 we report also the respective numerical values of the difference
in some points
.
![figure 5](http://media.springernature.com/lw685/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig5_HTML.jpg)
Figure 5
Remark.
We conclude this paper observing that, dividing by inequalities (1.10)-(1.11) and integrating them from
to
(
), we obtain the following new lower bounds for the ratios
and
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ31_HTML.gif)
For a survey on inequalities of the type (3.2) and (3.3) see [4].
In the following Tables 5 and 6 we confront the lower bounds (1.1)–(3.2) and (1.4)–(3.3), respectively, for different values of in the particular cases
and
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ32_HTML.gif)
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
).
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Acknowledgment
This work was sponsored by Ministero dell'Universitá e della Ricerca Scientifica Grant no. 2006090295.
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Laforgia, A., Natalini, P. Some Inequalities for Modified Bessel Functions. J Inequal Appl 2010, 253035 (2010). https://doi.org/10.1155/2010/253035
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DOI: https://doi.org/10.1155/2010/253035