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.