Abstract

If is a polynomial of degree , having no zeros in , then Aziz (1989) proved , where . In this paper, we consider a class of polynomial of degree , defined as and present certain generalizations of above inequality and some other well-known results.

1. Introduction and Statement of the Results

Let denote the space of all complex polynomials of degree . If , that is, , then according to the famous result of Bernstein (see [1]) and a simple deduction from the maximum modulus principle (see [2, page 346]) Both inequalities (1) and (2) are sharp and the equality holds if and only if has all its zeros at the origin.

If we restrict ourselves to the class of polynomial having no zeros in , then (1) can be sharpened. In fact, P. Erdös conjectured and later Lax [3] proved that if in , then It was shown by Frappier et al. [4, Theorem 8] that if , then and clearly (4) represents a refinement of (1), since the maximum of on may be larger than maximum of taken over th roots of unity as one can show by taking a simple example , .

In this connection, Aziz [5] improved inequality (4) by showing that if , then for every real and inequality (2) for where

Inequality (3) has also been improved by Aziz [5] by using the same argument as in inequality (5) and he proved that if and in , then for every real and also improved inequality (6) for every real and where is same as defined in (7).

Recently, Rather and Shah [6] considered the class of polynomials having no zeros in , and and improved the inequalities (8) and (9) of Aziz [5] by showing the following.

Theorem A. If , in , , and , then for every real where is defined in (7).

Theorem B. If , in , , and , then for every real and where is defined in (7).

In this paper, we first present the following result, which is a generalization of inequality (10) on class of polynomial of degree defined as , and having no zeros in , .

Theorem 1. If , in , , and , then for every real where is defined in (7).

On taking in Theorem 1, the following result has been obtained, which was shown by Rather and Shah [6].

Corollary 2. If , in and , then for every real where is defined in (7).

If we consider that some zeros of the polynomial are on , then and the following result has been obtained from Theorem 1.

Corollary 3. If and in , , then for every real where is defined in (7).

Remark 4. For , Corollary 3 reduces to inequality (8) due to Aziz [5].
Taking in inequality (14) of Corollary 3, we get the following result.

Corollary 5. If and in , , then for every real where is defined in (7).

Next result is the generalization of inequality (11) of Rather and Shah [6] for the polynomial and having no zeros in , .

Theorem 6. If , in , , and , then for every real and where is defined in (7).

Remark 7. For taking and , Theorem 6 reduces inequality (11) of Theorem B and on dividing both sides of inequality (16) by and taking , inequality (12) of Theorem 1 is obtained.

If we take and in inequality (16), we have the following inequality due to Rather and Shah [6].

Corollary 8. If , in and , then for every real and where is defined in (7).

If we take in inequality (16) of Theorem 6, we have the following result.

Corollary 9. If and in , , then for every and where is defined in (7).

Remark 10. From inequality (18), it is clear that Corollary 9 is a generalization of inequality (11). Also on dividing inequality (18) by and making we have inequality (13) of Corollary 2.
Another generalization of Theorem 6 has been obtained by considering the fact that some zeros of are on ; that is, at the following result has been found from Theorem 6.

Corollary 11. If and in , , then for every real and where is defined in (7).

Remark 12. For and , Corollary 11 reduces to inequality (9) due to Aziz [5] and dividing inequality (19) by and taking we have inequality (14) of Corollary 3.
An immediate consequence of Corollary 11 has to be obtained by taking .

Corollary 13. If and in , , then for every real and where is defined in (7).

The following result also gives a generalization of inequality (19) by taking in Corollary 11.

Corollary 14. If and in , , then for every real and where is defined in (7).

If we consider a class of polynomials , , , and having -fold zeros at origin and remaining zeros in , , then we prove the following.

Theorem 15. If , , is a polynomial of degree and having -fold zeros at origin and remaining zeros in , , then for every where

Remark 16. If there are no zeros at origin, that is if , then inequality (22) is reducing to inequality (14) of Corollary 3.
As an application of inequality (22), we obtained the following results by considering and , respectively, which are generalization of inequality (15) of Corollary 5 and inequality (8) due to Aziz [5] on -fold zeros, respectively.

Corollary 17. If and having -fold zeros at origin and remaining zeros in , , then for every where is defined in (23).

Corollary 18. If , having -fold zeros at origin and remaining zeros in , then for every where is defined in (23).

Inequality (22) of Theorem 15 can be improved by including . In this connection, instead of proving Theorem 15, we prove the following more general result.

Theorem 19. If , , is a polynomial of degree and having -fold zeros at origin and remaining zeros in , , then for every real and where is defined in (23).

By taking in inequality (26) of Theorem 15, we obtained the following.

Corollary 20. If , having -fold zeros at origin and remaining zeros in , , then for every real and where is defined in (22).

And for inequality (26) becomes as follows.

Corollary 21. If , having -fold zeros at origin and remaining zeros in , then for every real and where is defined in (22).

Remark 22. Corollaries 20 and 21 are generalization of inequalities (11) and (9) for -fold zeros of and for , Theorem 19 gives inequality (12) of Theorem 1.

2. Lemmas

For the proof of above results, the following lemmas are required. The very first lemma is due to Aziz [5].

Lemma 23. If , then for and for every real where is defined the same as in (7).

Next lemma is due to Aziz and Rather [7].

Lemma 24. If , in , , and , then for and where .

3. Proof of the Theorems

Proof of Theorem 1. By the hypothesis in , , dividing (30) by and taking in Lemma 24 we have, for , Since and for , ; therefore inequality (31) gives and using Lemma 23 in inequality (33), we have which completes the proof of Theorem 1.

Proof of Theorem 6. Applying the result (2) to the polynomial , which is of degree , and using (12), we obtained for and Hence for each , and , we have and, from inequality (35), we have which implies that, for and , Theorem 6 is completed.