## Abstract

By using the boundary Schwarz lemma, it was shown by Dubinin (J Math Sci 143:3069–3076, 2007) that if *P*(*z*) is a polynomial of degree n having all its zeros in \(|z| \le 1,\) then for all *z* on \(|z|=1\) for which \(P(z)\ne 0,\)

In this paper, by using simple techniques we generalize the above inequality, thereby give a simple proof of the above inequality. As an application of our result, we also obtain sharp refinements of some known results due to Malik (J Lond Math Soc 1:57–60, 1969), Aziz and Rather (Math Ineq Appl 1:231–238, 1998). These results take into account the size of the constant term and the leading coefficient of the polynomial *P*(*z*).

This is a preview of subscription content, access via your institution.

## References

- 1.
Aziz, A.: Inequalities for the derivative of a polynomial. Proc. Am. Math. Soc.

**89**(2), 259–266 (1983) - 2.
Aziz, A., Rather, N.A.: A refinement of a theorem of Paul Turán concerning polynomials. Math. Ineq. Appl.

**1**, 231–238 (1998) - 3.
Aziz, A., Rather, N.A.: Inequalities for the polar derivative of a polynomial with restricted zeros. Math. Balk.

**17**, 15–28 (2003) - 4.
Aziz, A., Rather, N.A.: Some Zygmund type L\(^q\) inequalities for polynomials. J. Math. Anal. Appl.

**289**, 14–29 (2004) - 5.
Dubinin, V.N.: Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros. J. Math. Sci.

**143**, 3069–3076 (2007) - 6.
Malik, M.A.: On the derivative of a polynomial. J. Lond. Math. Soc.

**1**, 57–60 (1969) - 7.
Marden, M.: Geometry of Polynomials, Mathematical Surveys, vol. 3. American Mathematical Society, Providence (1966)

- 8.
Milovanović, G.V., Mitrinović, D.S., Rassias, T.M.: Topics in Polynomials: Extremal Properties, Inequalities, Zeros. World scientific Publishing Co., Singapore (1994)

- 9.
Osserman, R.: A sharp Schwarz inequality on the boundary. Proc. Am. Math. Soc.

**128**(12), 3513–3517 (2000) - 10.
Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Clarendon Press, Oxford (2002)

- 11.
Turán, P.: Über die Ableitung von Polynomen. Compos. Math.

**7**, 89–95 (1939)

## Acknowledgements

We are grateful to the referees for their valuable suggestions, which have certainly enhanced the presentation of this paper.

## Author information

### Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Rather, N.A., Dar, I. & Iqbal, A. Some inequalities for polynomials with restricted zeros.
*Ann Univ Ferrara* (2020). https://doi.org/10.1007/s11565-020-00353-3

Received:

Accepted:

Published:

### Keywords

- Polynomials
- Inequalities
- Refinement

### Mathematics Subject Classification

- 26D10
- 41A17
- 30C15