- Open Access
Inequalities for a polynomial and its derivative
© Zireh; licensee Springer 2012
- Received: 14 December 2011
- Accepted: 13 September 2012
- Published: 2 October 2012
For a polynomial of degree n which has no zeros in , Liman et al. (Appl. Math. Comput. 218:949-955, 2011) established
for all with , , and . In this paper, we extend the above inequality for the polynomials having no zeros in , . Our result generalizes certain well-known polynomial inequalities.
MSC:30A10, 30C10, 30D15.
- maximum modulus
- restricted zeros
Inequality (1.1) is a famous result due to Bernstein , whereas inequality (1.2) is a simple consequence of the maximum modulus principle (see ). Both the above inequalities are sharp, and an equality in each holds for the polynomials having all their zeros at the origin.
Inequality (1.3) was conjectured by Erdös and later proved by Lax , whereas inequality (1.4) was proved by Ankeny and Rivlin , for which they made use of (1.3). Both these inequalities are also sharp, and an equality in each holds for polynomials having all their zeros on .
Both these inequalities are also sharp, and an equality in each holds for with .
Both these inequalities are also sharp, and an equality in each holds for polynomials having all their zeros on .
As an extension to inequality (1.11), we propose the following result.
If we take in Theorem 1, then inequality (1.12) reduces to (1.11).
Theorem 1 reduces to the following result by taking .
Dividing both sides of inequality (1.13) by and then making , we get the following generalization of inequality (1.9).
Taking in Theorem 1, we also obtain the following generalization of inequality (1.10).
If we take in Theorem 1, then we have the following consequence.
If we take in Corollary 1.4, then we get
Taking in Corollary 1.5, inequality (1.17) reduces to inequality (1.8).
For the proof of Theorem 1, we need the following lemmas. The first lemma is due to Aziz and Zargar .
lie in .
where and .
then , and with this choice of δ, we have for from (2.5). But this contradicts the fact that all the zeros of lie in . For β with , (2.6) follows by continuity. This completes the proof. □
If we take in Lemma 2.2, we have
If we take in Lemma 2.2, we get
Proof If we take in Lemma 2.2, the result follows. □
This, in conjunction with (2.13), gives the result. □
This gives the result. □
The proof follows some known ideas in the literature.
Considering inequalities (3.4) and (3.5) together gives the result. □
The author is grateful to the referees for the careful reading of the paper and for the helpful suggestions and comments. This research was supported by Shahrood University of Technology.
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