Some New Wilker and Generalized Lazarević Type Inequalities for Modified Bessel Functions
Département de Mathématiques ISSATK, University of Kairaouan, Kairaouan, TunisiaAbstract | |
1. | Introduction |
2. | Lemmas |
3. | Wilker and Lazarević Type Inequalities for Modified Bessel Functions |
4. | Concluding Remarks |
References |
Abstract
In this paper our aim is to deduce some new Wilker types inequalities for modied Bessel function of the first kind. In addition, a generalized Lazarević's inequality is established.
Keywords: The modified Bessel functions, Wilker type inequalities, Lazarević inequality
Received December 28, 2015; Revised November 19, 2016; Accepted November 29, 2016
Copyright © 2016 Science and Education Publishing. All Rights Reserved.Cite this article:
- Khaled Mehrez. Some New Wilker and Generalized Lazarević Type Inequalities for Modified Bessel Functions. Turkish Journal of Analysis and Number Theory. Vol. 4, No. 6, 2016, pp 168-171. http://pubs.sciepub.com/tjant/4/6/4
- Mehrez, Khaled. "Some New Wilker and Generalized Lazarević Type Inequalities for Modified Bessel Functions." Turkish Journal of Analysis and Number Theory 4.6 (2016): 168-171.
- Mehrez, K. (2016). Some New Wilker and Generalized Lazarević Type Inequalities for Modified Bessel Functions. Turkish Journal of Analysis and Number Theory, 4(6), 168-171.
- Mehrez, Khaled. "Some New Wilker and Generalized Lazarević Type Inequalities for Modified Bessel Functions." Turkish Journal of Analysis and Number Theory 4, no. 6 (2016): 168-171.
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1. Introduction
In the literature there are many inequalities satisfied by Bessel and modified Bessel functions of the first kind. Different types of inequalities for this functions are proved, like Turán types inequalities [4, 15], Jordan's type Inequalities [5], Redheffer's type inequalities [6, 8, 24], Huygens types inequalities [10, 11] and Frame's types inequalities [9], ...etc. This paper is a continuation of some inequalities for this functions. Wilker [19] proposed two open problems:
a. Prove that if then
![]() | (1) |
b. Find the largest constant c such that
![]() | (2) |
for
In [16], inequality (1) was proved, and the following inequality:
![]() |
where the constants and
are best possible, was also established.
Wilker-type inequalities (1) and (2) have attracted much interest of many mathematicians and have motivated a large number of research papers involving different proofs and various generalizations and improvements (cf. [3, 23] and the references cited therein).
In this paper, some new Wilker-type inequalities involving modified Bessel functions of the first kind are established. Moreover, we present a new proof of generalization of the Lazarević and Wilker-type inequalities proved by Baricz [2].
Let let us consider the function
defined by
![]() | (3) |
and is the modified Bessel function of the first kind defined by [[18], p. 77]
![]() | (4) |
It is worth mentioning that in particular we have,
![]() | (5) |
![]() | (6) |
![]() | (7) |
2. Lemmas
In order to establish our main results, we need several lemmas, which we present in this section.
Lemma 1. [14] Let and
be real numbers, and let the power series
and
be convergent for
If
for
and if
is strictly increasing (or decreasing) for
then the function
is strictly increasing (or decreasing) on
Lemma 2. [1, 8, 13] Let be two continuous functions which are differentiable on
Further, let
on
If
is increasing (or decreasing) on
then the functions
and
are also increasing (or decreasing) on
3. Wilker and Lazarević Type Inequalities for Modified Bessel Functions
The first aim of this paper is to prove the following inequalities.
Theorem 1. Let the following inequalities
![]() | (8) |
and
![]() | (9) |
holds for all
Proof. Let we define the function
on
by
![]() |
where and
By using the differentiation formula [[18], p. 79]
![]() | (10) |
we can easily show that
![]() | (11) |
and
![]() | (12) |
Thus
![]() | (13) |
Using the Cauchy product
![]() | (14) |
we get
![]() |
where
![]() | (15) |
and
![]() | (16) |
Now, we define the sequence for
thus
![]() |
We conclude that is increasing for
and
is also increasing on
by Lemma 1. Thus
![]() | (17) |
is increasing on by Lemma 2. Furthermore,
![]() |
from which follows the inequality (8) holds for all and
Finally, using the the arithmeticgeometric mean inequality and equality (9), we get
![]() | (18) |
Since the inequality (9) holds for all
and
So, the proof of Theorem 1 is complete.
In this theorem, we establish new inequalities of the Wilker type for modified Bessel function of the first kind.
Theorem 2. Let and
the following Wilker type inequality
![]() | (19) |
is valid. In particular, the following inequality
![]() | (20) |
hold for all
Proof. We define the function on
by
![]() |
The Mittag-Leffler expansion for the modified Bessel functions of first kind, which becomes [[7], Eq. 7.9.3]
![]() | (21) |
where are the positive zeros of the Bessel function
, and the differentiation formula (10) we have
![]() |
Thus implies that the function is increasing on
for all
and consequently
for all
So, the following inequality
![]() | (22) |
holds for all and
. By using the arithmetic-geometric mean inequality we obtain
![]() |
Finally, Observe that using (5) and (6) in particular for we obtain the inequality (20).
In the next theorem we present a generalization of the Lazarević and Wilker-type inequalities to modified Bessel functions of the first kind. The next result exist in [3]. We give an elementary proof.
Theorem 3. let and
Then, the following inequalities
![]() | (23) |
and
![]() | (24) |
holds if and only if
Proof. Let where
and
From the differentiation formula (10) we get
![]() |
where
![]() |
and
![]() |
We define the sequence by
for
thus implies that
![]() |
and consequently is decreasing for
so the function
is decreasing on
by Lemma 1. Thus
is decreasing on
by Lemma 2. Now,
![]() |
Therefore, (23) holds. From arithmetic-geometric mean inequality and inequality (23), we have
![]() |
and the proof of theorem is complete.
4. Concluding Remarks
1. In proof of Theorem 1, we can see that the following Turán type inequality
![]() | (25) |
holds for all and
This inequality is not new and it is actually equivalent to a very well-known Turán type inequality for the modified Bessel function of the first kind, firstly discovered by Thiruvenkatachar and Nanjundiah, see [17]
2. On the other hand, by using (5), (6) and (7) in particular for the Turán type inequality (25) becomes
![]() |
3. Inequality (23) is a natural generalization of the Lazarević inequality [[12], p. 207]
![]() | (26) |
where Recently, Zhu gives a new proof of the inequality (26) in [20] and extends the inequality (26) to the following result in [21] by: for
and
, then
![]() | (27) |
holds if and only if Moreover, in [22], Zhu gives another generalization of the inequality (26) as follows: for
or
then
![]() | (28) |
holds if and only if
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