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Research Article

Open Access Peer-reviewed

Frej Chouchene^{ }

Received January 01, 2018; Revised March 01, 2018; Accepted May 05, 2018

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.

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}.

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

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.

Thanks to the referee for careful reading and helpful comments.

[1] | Carver, W.B. and Oppenheim, A., Elementary problems and solutions: solutions: E 1277, Amer. Math. Monthly, 65(3). 206-209. 1958. | ||

In article | |||

[2] | Mitrinović, D.S., Analytic inequalities, Springer-Verlag, Berlin, 1970. | ||

In article | View Article | ||

[3] | Zhu, L., A solution of a problem of Oppenheim. Math. Inequal. Appl., 10(1). 57-61. 2007. | ||

In article | View Article | ||

[4] | Baricz, Á., Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math., 26(3). 279-293. 2008. | ||

In article | View Article | ||

[5] | Chouchene, F., Oppenheim's problem and related inequalities for Dunkl kernels, Math. Inequal. Appl., 17(1). 1-40. 2014. | ||

In article | View Article | ||

[6] | Baricz, Á. and Zhu, L., Extension of Oppenheim's problem to Bessel functions. J. Inequal. Appl., Art. ID 82038. 7 pp. 2007. | ||

In article | View Article | ||

[7] | Baricz, Á., Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, 1994. Springer-Verlag, Berlin, 2010. xiv+206 pp. | ||

In article | View Article | ||

[8] | Chettaoui, C. and Trimèche, K., New type Paley-Wiener theorems for the Dunkl transform on . Integral Transforms Spec. Funct., 14(2). 97-115. 2003. | ||

In article | View Article | ||

[9] | Dunkl, C.F., Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 311(1). 167-183. 1989. | ||

In article | View Article | ||

[10] | Dunkl, C.F., Integral kernels with reflection group invariance, Canad. J. Math., 43. 1213-1227. 1991. | ||

In article | View Article | ||

[11] | Dunkl, C.F., Hankel transforms associated to finite reflection groups, Contemp. Math., 138. 123-138. 1992. | ||

In article | View Article | ||

[12] | Dunkl, C.F., Intertwining operators and polynomials associated with the symmetric group, Monatsh. Math., 126. 181-209. 1998. | ||

In article | View Article | ||

[13] | Dunkl, C.F., Orthogonal polynomials of types A and B and related Calegero models, Commun. Math. Phys., 197. 451-487. 1998. | ||

In article | View Article | ||

[14] | Mourou, M.A., Transmutation operators associated with a Dunkl type differential-difference operator on the real line and certain of their applications, Integral Transforms Spec. Funct., 12(1). 77-88. 2001. | ||

In article | View Article | ||

[15] | Mourou, M.A. and Trimèche, K., Opérateurs de transmutation et théorème de Paley-Wiener associés à un opérateur aux dérivées et différences sur , C. R. Acad. Sci. Paris, Série I Math., 332. 397-400. 2001. | ||

In article | View Article | ||

[16] | Mourou, M.A. and Trimèche, K., Transmutation operators and Paley-Wiener theorem associated with a differential-difference operator on the real line, Anal. Appl., 1. 43-70. 2003. | ||

In article | View Article | ||

[17] | Rösler, M., Bessel-type signed hypergroups on . In: H. Heyer, A. Mukherjea (eds.). Probability measures on groups and related structures XI. Proceedings. Oberwolfach 1994, Singapore: World Scientific, 292-304. 1995. | ||

In article | |||

[18] | Watson, G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, UK, 1962. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2018 Frej Chouchene

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Frej Chouchene. Oppenheim's Problem and Some Inequalities Involving Bessel Functions and Dunkl Kernels. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 3, 2018, pp 61-71. http://pubs.sciepub.com/tjant/6/3/1

Chouchene, Frej. "Oppenheim's Problem and Some Inequalities Involving Bessel Functions and Dunkl Kernels." *Turkish Journal of Analysis and Number Theory* 6.3 (2018): 61-71.

Chouchene, F. (2018). Oppenheim's Problem and Some Inequalities Involving Bessel Functions and Dunkl Kernels. *Turkish Journal of Analysis and Number Theory*, *6*(3), 61-71.

Chouchene, Frej. "Oppenheim's Problem and Some Inequalities Involving Bessel Functions and Dunkl Kernels." *Turkish Journal of Analysis and Number Theory* 6, no. 3 (2018): 61-71.

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[1] | Carver, W.B. and Oppenheim, A., Elementary problems and solutions: solutions: E 1277, Amer. Math. Monthly, 65(3). 206-209. 1958. | ||

In article | |||

[2] | Mitrinović, D.S., Analytic inequalities, Springer-Verlag, Berlin, 1970. | ||

In article | View Article | ||

[3] | Zhu, L., A solution of a problem of Oppenheim. Math. Inequal. Appl., 10(1). 57-61. 2007. | ||

In article | View Article | ||

[4] | Baricz, Á., Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math., 26(3). 279-293. 2008. | ||

In article | View Article | ||

[5] | Chouchene, F., Oppenheim's problem and related inequalities for Dunkl kernels, Math. Inequal. Appl., 17(1). 1-40. 2014. | ||

In article | View Article | ||

[6] | Baricz, Á. and Zhu, L., Extension of Oppenheim's problem to Bessel functions. J. Inequal. Appl., Art. ID 82038. 7 pp. 2007. | ||

In article | View Article | ||

[7] | Baricz, Á., Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, 1994. Springer-Verlag, Berlin, 2010. xiv+206 pp. | ||

In article | View Article | ||

[8] | Chettaoui, C. and Trimèche, K., New type Paley-Wiener theorems for the Dunkl transform on . Integral Transforms Spec. Funct., 14(2). 97-115. 2003. | ||

In article | View Article | ||

[9] | Dunkl, C.F., Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 311(1). 167-183. 1989. | ||

In article | View Article | ||

[10] | Dunkl, C.F., Integral kernels with reflection group invariance, Canad. J. Math., 43. 1213-1227. 1991. | ||

In article | View Article | ||

[11] | Dunkl, C.F., Hankel transforms associated to finite reflection groups, Contemp. Math., 138. 123-138. 1992. | ||

In article | View Article | ||

[12] | Dunkl, C.F., Intertwining operators and polynomials associated with the symmetric group, Monatsh. Math., 126. 181-209. 1998. | ||

In article | View Article | ||

[13] | Dunkl, C.F., Orthogonal polynomials of types A and B and related Calegero models, Commun. Math. Phys., 197. 451-487. 1998. | ||

In article | View Article | ||

[14] | Mourou, M.A., Transmutation operators associated with a Dunkl type differential-difference operator on the real line and certain of their applications, Integral Transforms Spec. Funct., 12(1). 77-88. 2001. | ||

In article | View Article | ||

[15] | Mourou, M.A. and Trimèche, K., Opérateurs de transmutation et théorème de Paley-Wiener associés à un opérateur aux dérivées et différences sur , C. R. Acad. Sci. Paris, Série I Math., 332. 397-400. 2001. | ||

In article | View Article | ||

[16] | Mourou, M.A. and Trimèche, K., Transmutation operators and Paley-Wiener theorem associated with a differential-difference operator on the real line, Anal. Appl., 1. 43-70. 2003. | ||

In article | View Article | ||

[17] | Rösler, M., Bessel-type signed hypergroups on . In: H. Heyer, A. Mukherjea (eds.). Probability measures on groups and related structures XI. Proceedings. Oberwolfach 1994, Singapore: World Scientific, 292-304. 1995. | ||

In article | |||

[18] | Watson, G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, UK, 1962. | ||

In article | |||