Abstract

In this paper, we establish some inequalities related to Oppenheim's problem for the real and imaginary parts of Dunkl kernels In order to prove our main results, we present some new inequalities involving Bessel functions of the first kind. Refinements of inequalities for Bessel functions are also given.

1. Introduction

We consider the Oppenheim's problem: What are the best possible constants such that

hold for all

In 1958, a partial solution is given in ^{1, 2}: If and , then

Next, L. Zhu solved completely this problem for trigonometric functions, see ³. Since the cosine and sine functions are particular cases of Bessel functions, then it is natural to generalize some formulas and inequalities involving these elementary functions to Bessel functions. The extension of the Oppenheim's problem to Bessel and modified Bessel functions was first considered by Á. Baricz in ⁴. Recently, we established in ⁵ some inequalities related to this type of problem for Dunkl kernels by answering to the following question: What are, for the best possible constants such that

hold for all

Some new inequalities involving modified Bessel functions have also been improved.

Our aim is to solve the analogues of the Oppenheim's problem for the real and imaginary parts of Dunkl kernels In the beginning, we present some new inequalities related to this problem for trigonometric functions. These inequalities and Sonine integral formula for Bessel functions allow us to get a new version of the solution of this type of problem for Bessel functions Next, by using again Sonine integral formula for Bessel functions, we solve the Oppenheim's problem for the imaginary parts of Dunkl kernels At the end of this paper, we give refinements of inequalities for Bessel functions . More precisely, in view of the inequalities given by Á. Baricz in ^{4, 6}, we prove that if and , then for all , we have

and

If then for all we have

where and are as in Theorem of ⁶.

2. Dunkl Kernels

In this section, we take

Definition 2.1. Let We call Dunkl kernel the function defined by

where is the normalized Bessel function of index given by

with for all , we have

(1)

For more details of these functions, we can see ^{7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}. The following results of this section are proved in ⁵.

Proposition 2.2. For all and , we have

1) .

2) ,

3)

4)

5)

6)

7) .

In particular, if then

8) .

Proposition 2.3. The Dunkl kernel is the unique entire solution on of the following equation:

where is the Dunkl operator on of index associated to the reflections group and given by

Remarks 2.4. If , then we have

1)

2)

3)

4)

where (resp. ) is the even (resp. odd) part of

In particular, for all , we have

where is the normalized Bessel function of index .

Proposition 2.5. For all and , we have

1) .

2) , with for all , we have

(2)

where is given by (1).

3)

4)

5)

6)

7) The function is the unique entire solution on of the following equation:

Examples 2.6. For all , we have

1)

2)

3)

where is the normalized modified Bessel function of index given by

with is given by (1) .

Remarks 2.7.

1) The function is the unique entire solution on of the following equation:

2) The function is the unique entire solution on of the following equation:

where is the Bessel operator on given by

Proposition 2.8. For all and , we have

1) .

2) .

3) .

4) .

Proposition 2.9. For all and , we have

1)

2) If , then

3) If then

3. Inequalities Related to Oppenheim's Problem

We begin to find the best possible constants such that

(3)

hold for all ,

The solution of this problem can be stated in the following theorem:

Theorem 3.1

1) For all and , we have

2) For all and , we have

3) For all and , we have

4) For all and , we have

5) For all and , we have

where and are respectively given by (4), (5) and (6).

With the aid of Sonine integral formula for Bessel functions we get the following theorem:

Theorem 3.2 Let .

1) For all and , we have

2) For all and , we have

3) For all and , we have

where and are respectively given by (4), (5) and (6).

Thus, by choosing in Theorem 3.2, we obtain the following interesting result.

Corollary 3.3

1) For all and , we have

2) For all , we have

3) For all and , we have

4) For all , we have

5) For all and , we have

where and are respectively given by (4), (5) and (6).

Now, for , we are going to find the best possible constants such that

hold for all

The solution of this problem can be stated in the following theorem:

Theorem 3.4 Let .

1) For all and , we have

2) For all and , we have

3) For all and , we have

4) For all and , we have

5) For all and , we have

6) For all and , we have

where and are respectively given by (4), (5) and (6).

In order to solve Problem (3), we present the following propositions:

Proposition 3.5

1) For all and , we have

2) For all and , we have

3) For all and , we have

where for ,

(4)

(5)

Proposition 3.6

1) For all and , we have

2) For all and , we have

3) For all and , we have

4) For all and , we have

where and are respectively given by (4) and (5).

Proposition 3.7

1) For all , we have

2) For all and , we have

3) For all and , we have

4) For all and , we have

where and are respectively given by (4), (5) and for ,

(6)

The study of the Bessel function gives the following proposition:

Proposition 3.8 The function is even on strictly increasing on , strictly decreasing on , and satisfies

1)

2) .

3) .

In view of ¹⁸ we deduce the following Sonine integral formulas:

Proposition 3.9 For all and , we have

1)

2)

1) Let . We have

a) If , then

b) If , then

c) If , then

where and

2) For all and , we have

3) For all and , we have

4) For all and , we have

5) For all and , we have

6) For all , we have

7) For all and , we have

8) For all and , we have

In view of the inequalities for Bessel functions given by Á. Baricz in ^{4, 6}, we can show the refinements of these inequalities as follows.

9) Let and

We have

a) .

b) .

c) .

d) .

10) Let and . We have

a) .

b) .

c)

d)

11) Let and We have

a)

b) .

c)

d)

12) Let , and . We have

a) .

b) .

c) .

d)

13) Let and . We have

a) .

b) .

c)

d)

14) Let . Á. Baricz has proved in ^{4, 6} that for all , we have

where and are as in Theorem 1.1 of ⁶.

a) If

then , and for all , we have

and

b) If , then for all

we have

Proof of Proposition 3.5: Let and

We have .

if and only if

, where

.

is strictly increasing on ,

is strictly increasing on ,

If , then is strictly decreasing on ,

If , then there exists such that is strictly increasing on and strictly decreasing on ,

where

is strictly decreasing on and strictly increasing on ,

Proof of Proposition 3.6: Let , and

We have .

if and only if , where is given by (5).

If , then is strictly increasing on and ,

If , then is strictly increasing on ,

If , then there exist such that

is strictly decreasing on and , and strictly increasing on ,

where is given by (4).

If , then there exist such that , is strictly increasing on and , and strictly decreasing on and ,

Proof of Proposition 3.7: Let , and

We have

if and only if

, where is given by (7). is strictly increasing on ,

If , then is strictly increasing on

If , then is strictly increasing on ,

If , then is strictly increasing on ,

If , then there exists such that .

is strictly decreasing on and strictly increasing on ,

where is given by (4).

Proof of Proposition 3.8: Let and

We have ,

is strictly decreasing on ,

is even on and strictly increasing on and strictly decreasing on ,

Proof of Theorem 3.1: We get the result from Propositions 3.5, 3.6, 3.7 and Remarks 1, …,, 8.

Proof of Theorem 3.2: Let ,

and .

If , then by using Proposition 3.9 and

we deduce

Theorem 3.1 finishes the proof.

Proof of Theorem 3.4: Let ,

and .

If , then by using Proposition 3.9 and

we deduce

Theorem 3.2 with finishes the proof.

Acknowledgements

Thanks to the referee for careful reading and helpful comments.

References