ISRN Mathematical Analysis

Volume 2014 (2014), Article ID 272405, 8 pages

http://dx.doi.org/10.1155/2014/272405

Research Article

Some Inequalities for Polynomials Not Vanishing inside a Circle

Abdullah Mir and Bilal Dar

Department of Mathematics, University of Kashmir, Hazratbal Srinagar 190006, India

Received 18 January 2014; Accepted 16 February 2014; Published 19 May 2014

Academic Editors: K. A. Lurie and M. A. McKibben

Copyright © 2014 Abdullah Mir and Bilal Dar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

If is a polynomial of degree having no zeros in , then it is known that, for all with , , , and , . In this paper, we will prove a result which not only generalizes the above inequality but also generalize and refines the various results pertaining to the norm of . We will also prove a result which extends and refines a result of Boas Jr. and Rahman (1962). Also we will see that our results lead to some striking conclusions giving refinements and generalizations of other well-known results.

1. Introduction

For an th degree polynomial , define

If is a polynomial of degree , then

Inequality (2) is an immediate consequence of a famous result due to Bernstein [1] on the derivative of a trigonometric polynomial, whereas inequality (3) is a simple deduction from the maximum modulus principle [2, page 346].

Restricting ourselves to a class of polynomials having no zero in , the inequalities (2) and (3) can be, respectively, replaced by

Inequality (4) was conjectured by Erdös and later verified by Lax [3], whereas Ankeny and Rivilin [4] used (4) to prove (5). Inequalities (4) and (5) were further improved in [5], where under the same hypothesis it was shown that

Both inequalities (4) and (5) were generalized by Jain [6], who proved that if in , then for every with ,

for and .

Further, Aziz and Rather [7] generalised inequality (8) by proving that if is a polynomial of degree which does not vanish in , then for all with , and ,

The following result given in [8] provides a generalization of inequality (9) which interalia yields a compact generalization of inequality (8).

Theorem A. If is a polynomial of degree which does not vanish in , then for all with , , and ,

Further the following result given in [8] provides a refinement of Theorem A which among other results provides a compact generalization of inequalities (6) and (7) as well.

Theorem B. If is a polynomial of degree which does not vanish in , then for all with , , and ,

Recently, the dependence of on , was investigated in [9] for arbitrary complex numbers , with , , , and the following compact generalization of Theorem A was proved.

Theorem C. If is a polynomial of degree which does not vanish in , then for all with , , and ,

In this paper, we first prove the following more general result analogous to Theorem C which not only generalizes Theorem B to the -norm of for every but also leads to some striking conclusions giving refinements and generalizations of other well-known results.

Theorem 1. If is a polynomial of degree which does not vanish in , then for every with , , , , and ,

The result is best possible and equality holds in (13) for the polynomial .

A variety of interesting results can be easily deduced from Theorem 1; here we mention few of them.

For , Theorem 1 reduces to Theorem C. Also for , Theorem 1 reduces to a result of Aziz and Rather [10].

Further, on applying Minkowski’s inequality on the right hand side of (13), we obtain, for ,

Now by taking and in the above inequality and then letting , we get inequality (9).

Also by taking in inequality (14) and making , we obtain Theorem A.

The following corollary which is a compact generalization of (7) follows from Theorem 1 by taking .

Corollary 2. If is a polynomial of degree which does not vanish in , then for every with , , , and ,

The result is best possible and equality holds in (15) for the polynomial .

Remark 3. For , Corollary 2 reduces to a result of Aziz and Rather [11]. For and , Corollary 2 reduces to a result of Aziz and Rather [12]. Again for and , we get a result recently proved by Rather [13, Theorem 1.1].

Finally, as an application of Theorem 1, we prove the following generalization and refinement of a result of Boas Jr. and Rahman [14] for .

Theorem 4. If is a polynomial of degree which does not vanish in , then for every with , and , ,

For , Theorem 4 reduces to a result of Boas Jr. and Rahman [14] for . Also for , Theorem 4 reduces to the following corollary which is a compact generalization of inequality (7) due to Aziz and Dawood [5, Theorem 2] to norm.

Corollary 5. If is a polynomial of degree which does not vanish in , then for every with , and ,

2. Lemmas

For the proof of these theorems we need the following lemmas.

Lemma 6. If is a polynomial of degree having all its zeros in , then for all with , and ,

Lemma 7. If is a polynomial of degree which does not vanish in , then for all with , , and ,

where .

The above two lemmas are proved in [8].

Lemma 8. If is a polynomial of degree which does not vanish in , then for all with , and ,

where .

Proof of Lemma  8. If has a zero on , then and the result follows from Lemma 7. Therefore, we assume that has all its zeros in , so that . Now for any with , we have , for . By Rouche’s theorem, the polynomial has no zero in . If , then the polynomial has all its zeros in and also on . Therefore by Lemma 7, for all with and , we have

Equivalently,

Now choosing the argument of on the right hand side of (22) such that

which is possible by Lemma 6 and the fact that , we get, for ,

Now, if in (24) we make , we get

which is inequality (20) and that proves Lemma 8 completely.

Lemma 9. If is a polynomial of degree having no zeros in , then for every with , and , and real,

The above lemma is proved in [9].

Lemma 10. If , , and are nonnegative real numbers such that , then for every real number ,

The above lemma is due to Aziz and Rather [15].

Lemma 11. If is a polynomial of degree , then for every and ,

The above lemma is a simple consequence of a result of Hardy [16].

3. Proof of Theorems

Proof of Theorem 1. Since in , therefore, by Lemma 8, for each , and for all with , and , we have

where and .

This implies that

or

Taking

in Lemma 10 and noting by (31) that , we get, for every real ,

This implies, for each , that

where

Integrating both sides of (34) with respect to from 0 to , we get with the help of Lemma 9, for each and real,

Now for every real and , we have

which implies, for ,

If , we take ; then from (31), we have ; hence

For , this inequality is trivially true. Using this in (36), we conclude that for all with , , , , and ,

Now using the fact that for every complex number with ,

the desired result follows immediately by using (41) in (40).

This completes the proof of Theorem 1.

Proof of the Theorem 4. Since Theorem 1 holds for every with , taking in particular , we get

Now by Minkowski’s inequality, we have, for every ,

Using inequalities (28) (of Lemma 11) and (42) in (43), we get, for every with , , and ,

which is inequality (16) and this completes the proof of Theorem 4.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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