Abstract

In this paper we establish Turán type inequalities for Jacobi-Dunkl polynomials. These complete an earlier result proven by G. Gasper in [1], for Jacobi polynomials.

1. Introduction

The famous result of P. Turán ², established in 1950, is the following inequality:

where is the Legendre polynomial of degree This inequality has found much attention and several authors have deduced analogous results for other classical polynomials and special functions. For details we refer the interested reader to ^{1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} and their references. There are many applications of these results, for example in numerical integration, information theory, economic theory and biophysics (see ¹⁸).

Legendre polynomials belong to the class of Jacobi polynomials, which are studied in detail in ^{19, 20, 21}. For and let

where

is the normalized Jacobi polynomial of degree such that .

G. Gasper has proved in ¹ the following inequality:

with equality only for .

This shows that for all we have if and only if .

In the present paper our aim is to establish some inequalities of Turán type for Jacobi-Dunkl polynomials defined on by

if , and , with

We recall in the second section, some basic properties of Jacobi and Jacobi-Dunkl polynomials. We also present several explicit forms of these polynomials. Finally, we establish, in the last section, some inequalities of Turán type involving and (Theorems 3.4, 3.7, 3.9). More precisely,

with equality only for .

Then, we deduce that

if and only if .

In the sequel, we take and we denote by . We drop the exponents when there is no confusion.

2. Preliminaries

In this section we recall some properties for Jacobi and Jacobi-Dunkl polynomials useful to get inequalities of Turán type which occur in the study of certain entire functions. These inequalities give an idea about the size of the gap, estimates and monotonicity properties of some special functions.

2.1. Properties of Jacobi Polynomials

We consider the Jacobi polynomial defined on by

where is the normalized Jacobi polynomial such that . According t19,22], the ’s satisfy the following relations:

(1)

(2)

(3)

For all , the Jacobi polynomial is a solution on of the following differential equation: , where

In hypergeometric form, we have

where is the gaussian hypergeometric function given by

with

The derivative of is given by

In particular, In the Gegenbauer case (), we have

(4)

Explicit formulas of are given, for all and , by

(5)

where

(6)

if , and , and

(7)

The orthogonality property is given by

where

(8)

if and

We also have, for all and , the following recurrence formula:

Examples 2.1 For all and , we have

1)

2) ,

3) ,

2.1. Properties of Jacobi-Dunkl Polynomials

We take for all and , and for all .

For all , the Jacobi-Dunkl polynomial is defined on by

(9)

if , and ,

with (see ²²).

Remarks 2.2

1)

2) In ²², the author has studied the harmonic analysis associated with these polynomials.

Proposition 2.3 For all and , we have

1) ,

2) .

3)

4)

(10)

5)

(11)

6)

(12)

Proof: We use (1), (9), (4), (2) and (3) to get the above formulas.

For all is the unique solution on of the differential-difference equation

where is the Jacobi-Dunkl operator defined on by

In the following proposition we give explicit forms of

Proposition 2.4 For all and , we have

1)

2)

where and , are respectively given by (6) and (7).

Proof: We obtain the above formulas from (9) and (5).

The orthogonality property is given in ²² by , with

and , where , is given by (8).

Examples 2.5 For all and , we have

1) ,

2)

3) .

4)

5)

6)

3. Turán Type Inequalities

3.1. General Case

G. Gasper has proved in ¹ the following proposition:

Proposition 3.1 For all , we have

1)

with equality only for

2) if and only if .

In the sequel if this subsection, let be an integer not belonging to .

Lemma 3.2

1) .

2) .

Proof: Note that and

If , then

If , then and

If , then

and .

Lemma 3.3

Proof: Let and .

If , then .

If , then and

If , then and

Hence and

The following theorem gives a Turán type inequality for .

Theorem 3.4

with equality only for .

Proof: Let .

Proposition 3.1 and Lemma 3.2 give the above inequality.

Corollary 3.5

if and only if .

Remark 3.6

We establish now the following Turán type inequality for the Jacobi-Dunkl polynomials

Theorem 3.7

with equality only for .

Proof: Let .

From Proposition 3.1 it is obvious that

and

Note that and, by Lemma 3.2, .

To finish the proof we use Lemma 3.3.

Corollary 3.8

if and only if .

Another inequality of Turán type is given in the following theorem:

Theorem 3.9

with equality only for

Proof: Let

Then we complete the proof as in that of Theorem 3.7.

Corollary 3.10

if and only if .

Bu using (3) and (2), Proposition 3.1 may be rewritten in the following form:

Proposition 3.11 For all , we have

1) ,

with equality only for .

2) ,

if and only if .

From (12) and (11), we deduce the following proposition which gives again Turán type inequalities for .

Proposition 3.12

1)

with equality only for

2)

if and only if .

It is obvious that Theorem 3.7 and Corollary 3.8 may be rewritten in the following proposition:

Proposition 3.13

1) ,

with equality only for .

2)

if and only if .

Also, Theorem 3.9 and Corollary 3.10 may be rewritten in the following form:

Proposition 3.14

1) ,

with equality only for

2)

if and only if .

3.2. Special Cases

From (4), Proposition 3.1, and Proposition 3.11, it is clear that the Gegenbauer polynomials , satisfy the following inequalities:

Proposition 3.15 For all , we have

1) ,

with equality only for .

2) ,

if and only if .

3) ,

with equality only for .

4)

if and only if .

5) ,

with equality only for .

6) ,

if and only if .

7) ,

with equality only for .

8)

if and only if .

From (10), Theorem 3.4, Corollary 3.5, and Proposition 3.12, we deduce the following inequalities related to :

Proposition 3.16 For all , we have

1) ,

with equality only for .

2) ,

if and only if .

3) ,

with equality only for .

4)

if and only if .

By using (10), Theorem 3.7, Corollary 3.8, and Proposition 3.13, we can show that the Gegenbauer-Dunkl polynomials satisfy the following inequalities:

Proposition 3.17 For all , we have

1) ,

with equality only for .

2) ,

if and only if .

3)

4)

if and only if .

Also, (10), Theorem 3.9, Corollary 3.10, and Proposition 3.14 give the following proposition:

Proposition 3.18 For all , we have

1) ,

with equality only for .

2)

if and only if .

3)

with equality only for .

4)

if and only if .

Examples 3.19

1)If we take in Proposition 3.1, one can show that

2) If we take and in Proposition 3.1, one can show that

The study of the first terms of the Jacobi-Dunkl polynomials gives the following inequalities:

Proposition 3.20

with equality only for .

Proof:

Remark 3.21 The members of the inequality, given in Corollary 3.5, take the value zero when and .

Proposition 3.22

1) If , then

with equality only for or ( and ).

2) If , let

(13)

then

a) ,

b) ,

c) ,

Proof:

The derivative of the function

is Then Moreover, and when . Thus we get the sign of which finishes the proof.

Corollary 3.23

1) If , then

with equality only for or ( and ).

2) If , then

a) ,

b) ,

c) ,

where is given by (13).

Proposition 3.24 Let .

1) If , then

with equality only for or ( and ).

2) If , let

(14)

then

a) ,

b) ,

c) ,

Proof:

The derivative of the function

is If then Moreover, and . Thus we get the sign of which finishes the proof.

Remark 3.25

Corollary 3.26

1) If , then

with equality only for or ( and ).

2) If , then

a) .

b) .

c) ,

where is given by (14).

Proposition 3.27 Let .

1) If , then

with equality only for or ( and ).

2) If , let

(15)

then

a) ,

b) ,

c) ,

Proof:

The derivative of the function

is If , then . Moreover, and . Thus we get the sign of which finishes the proof.

Corollary 3.28

1) If , then

with equality only for or ( and ).

2) If , then

a) ,

b) ,

c) ,

where is given by (15).

Acknowledgements

Thanks to the referee for careful reading and helpful comments.

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