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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