Open Access

Some Inequalities for Modified Bessel Functions

Journal of Inequalities and Applications20102010:253035

DOI: 10.1155/2010/253035

Received: 15 October 2009

Accepted: 28 December 2009

Published: 24 January 2010

Abstract

We denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq2_HTML.gif the Bessel functions of the first and third kinds, respectively. Motivated by the relevance of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq4_HTML.gif , in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector (1984), we establish new inequalities for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq5_HTML.gif . The results are based on the recurrence relations for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq7_HTML.gif and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq8_HTML.gif .

1. Introduction

Inequalities for modified Bessel functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq10_HTML.gif have been established by many authors. For example, Bordelon [1] and Ross [2] proved the bounds

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ1_HTML.gif
(1.1)

The lower bound was also proved by Laforgia [3] for larger domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq11_HTML.gif . In [3] also the following bounds:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ4_HTML.gif
(1.4)

have been established; see also [4]

In this paper we continue our investigations on new inequalities for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq13_HTML.gif , but now our results refer not only to a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq14_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq15_HTML.gif at two different points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq17_HTML.gif , as in (1.1)–(1.4), but to two functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq19_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq21_HTML.gif ) and, more precisely, to the ratio https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq22_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq23_HTML.gif . This kind of ratios appears often in applied sciences. Recently, for example, the ratio https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq24_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq25_HTML.gif -function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ5_HTML.gif
(1.5)

(see also [6]). This generalized function and the classical one, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq26_HTML.gif , 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]

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ7_HTML.gif
(1.7)

and the Turán-type inequalities

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ8_HTML.gif
(1.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ9_HTML.gif
(1.9)

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

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ10_HTML.gif
(1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ11_HTML.gif
(1.11)

The results are given by the following theorems.

Theorem 1.1.

For real https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq27_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq28_HTML.gif be the modified Bessel function of the first kind and order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq29_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ12_HTML.gif
(1.12)

In particular, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq30_HTML.gif , the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq31_HTML.gif holds also true.

Theorem.

For real https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq32_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq33_HTML.gif be the modified Bessel function of the third kind and order https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq34_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ13_HTML.gif
(1.13)

In particular, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq35_HTML.gif , the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq36_HTML.gif holds also true.

2. The Proofs

Proof of Theorem 1.1.

The upper bound for the ratio https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq37_HTML.gif follows from the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ14_HTML.gif
(2.1)

proved by Soni for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq38_HTML.gif [16], and extended by Näsell to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq39_HTML.gif [17].

To prove the lower bound in (1.12), we substitute the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq40_HTML.gif given by (1.6) in the Turán-type inequality (1.8). We get, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq41_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ15_HTML.gif
(2.2)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ16_HTML.gif
(2.3)
We denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq42_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq43_HTML.gif and observe that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq44_HTML.gif , by (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq45_HTML.gif . With this notation (2.3) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ17_HTML.gif
(2.4)
which gives, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq46_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ18_HTML.gif
(2.5)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ19_HTML.gif
(2.6)

which is the desired result.

Remark.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq47_HTML.gif , Jones [18] proved stronger result than (2.1) that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq48_HTML.gif decreases with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq49_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq50_HTML.gif .

Proof of Theorem 1.2.

The proof is similar to the one used to prove Theorem 1.1. By
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ20_HTML.gif
(2.7)

we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq51_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq52_HTML.gif .

We substitute the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq53_HTML.gif given by (1.7) in (1.9). We get

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ21_HTML.gif
(2.8)
or, equivalently
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ22_HTML.gif
(2.9)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ23_HTML.gif
(2.10)
Finally, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ24_HTML.gif
(2.11)

which is the desired result (1.13).

Remark.

By means the integral formula [11, page 181]

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ25_HTML.gif
(2.12)
follows immediately the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ26_HTML.gif
(2.13)
and consequently
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ27_HTML.gif
(2.14)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq54_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq55_HTML.gif , only in this case the above upper bound for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq56_HTML.gif improves the (1.13) one.

Remark.

We observe that by Theorem 1.1 we obtain an upper bound for the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq58_HTML.gif . The investigations of the properties of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq59_HTML.gif are motivated by some problems of finite elasticity [19, 20]. By (1.12) we find
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ28_HTML.gif
(2.15)

in particular, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq60_HTML.gif , we also have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq61_HTML.gif .

3. Numerical Considerations

Baricz obtained, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq62_HTML.gif , the following similar lower bound for the ratio https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq63_HTML.gif (see [5, formula (5)])

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ29_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq64_HTML.gif is the unique simple positive root of the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq65_HTML.gif . Inequality (3.1) is reversed when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq66_HTML.gif . It is possible to prove that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq67_HTML.gif , our lower bound in (1.12) for the ratio https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq68_HTML.gif provides an improvement of (3.1).

Proposition.

Let be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq69_HTML.gif . Putting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq71_HTML.gif , one has https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq72_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq73_HTML.gif .

Proof.

From the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq74_HTML.gif we obtain, by simple calculations, the following one https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq75_HTML.gif which is satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq76_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq77_HTML.gif .

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

Example.

In the first case we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq78_HTML.gif . In Figure 1 we report the graphics of the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq79_HTML.gif (solid line) and the respective lower bounds https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq80_HTML.gif (short dashed line) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq81_HTML.gif (long dashed line) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq82_HTML.gif .

In Table 1 we report also the respective numerical values of the differences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq84_HTML.gif in some points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq85_HTML.gif .

Table 1

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq86_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq87_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq88_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq89_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq90_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq91_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq92_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq93_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq94_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq95_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq96_HTML.gif

https://static-content.springer.com/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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq97_HTML.gif and, in particular, for these values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq98_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq99_HTML.gif . See, for example, in Figure 2 the graphics of the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq100_HTML.gif (solid line) and the respective lower bounds https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq101_HTML.gif (short dashed line) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq102_HTML.gif (long dashed line) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq103_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq104_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig2_HTML.jpg

Figure 2

Example.

In this case we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq105_HTML.gif , then we report, in Figure 3, the graphics of the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq106_HTML.gif (solid line) and the respective lower bounds https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq107_HTML.gif (short dashed line) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq108_HTML.gif .In Table 3 we report also the respective numerical values of the differences https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq109_HTML.gif in some points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq110_HTML.gif .
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig3_HTML.jpg

Figure 3

Example.

Also in this case we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq111_HTML.gif . In Figure 4 we report the graphics of the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq112_HTML.gif (solid line) and the respective upper bound https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq113_HTML.gif (short dashed line) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq114_HTML.gif .In Table 2, we report also the respective numerical values of the difference https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq115_HTML.gif in some points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq116_HTML.gif .

Table 2

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq117_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq118_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq119_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq120_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq121_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq122_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq123_HTML.gif

Table 3

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq124_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq125_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq126_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq127_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq128_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq129_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq130_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Fig4_HTML.jpg

Figure 4

Example.

In this last case we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq131_HTML.gif . In Figure 5 we report the graphics of the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq132_HTML.gif (solid line) and the respective upper bound https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq133_HTML.gif (short dashed line) on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq134_HTML.gif . In Table 4 we report also the respective numerical values of the difference https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq135_HTML.gif in some points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq136_HTML.gif .

Table 4

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq137_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq138_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq139_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq140_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq141_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq142_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq143_HTML.gif

https://static-content.springer.com/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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq144_HTML.gif inequalities (1.10)-(1.11) and integrating them from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq145_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq146_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq147_HTML.gif ), we obtain the following new lower bounds for the ratios https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq148_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq149_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ30_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ31_HTML.gif
(3.3)

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq150_HTML.gif in the particular cases https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq152_HTML.gif . Let

Table 5

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq153_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq154_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq155_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq156_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq157_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq158_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq159_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq160_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq161_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq162_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq163_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq164_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq165_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq166_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq167_HTML.gif

Table 6

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq168_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq169_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq170_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq171_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq172_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq173_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq174_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq175_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq176_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq177_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq178_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq179_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq180_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq181_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq182_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq183_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq184_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_Equ32_HTML.gif
(3.4)

then we have Tables 5 and 6.

By the values reported on Table 5 it seems that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq185_HTML.gif is a lower bound much more stringent with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq186_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq187_HTML.gif (moreover we recall that (3.2) holds true also for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq188_HTML.gif ), while by the values reported on Table 6 it seems that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq189_HTML.gif is a lower bound more stringent with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq190_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq191_HTML.gif (but we recall that (3.3) holds true also for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F253035/MediaObjects/13660_2009_Article_2098_IEq193_HTML.gif ).

Declarations

Acknowledgment

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

Authors’ Affiliations

(1)
Department of Mathematics, Rome Tre University

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Copyright

© A. Laforgia and P. Natalini. 2010

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