# Inequalities for the derivative of polynomials with restricted zeros

## Main Article Content

## Abstract

For a polynomial $\mathit{P(z)=\sum_{\nu =0}^{n}a_{\nu}z^{\nu}}$ of degree $\mathit{n}$ having all its zeros in $\mathit{|z|\leq k,k \geq 1}$, it was shown by Rather and Dar \cite{1} that

$$\max_{|z|=1} |P^{\prime}(z)|\geq \frac{1}{1+k^n}\bigg(n+\frac{k^n|a_n|-|a_0|}{k^n|a_n|+|a_0|}\bigg)\max_{|z|=1}|P(z)|.$$

In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Tur\'{a}n-type inequalities.

## Article Details

## References

[1] Abdul Aziz, Inequalities for the derivative of a polynomial, Proc. Amer. Math. Soc. 89 (2) (1983), 259–266. Google Scholar

[2] A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivatives, J. Approx. Theory 54 (1998), 306–313. Google Scholar

[3] A. Aziz and N. A. Rather, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl. 289 (2004), 14–29. Google Scholar

[4] A. Aziz and N. A. Rather, Inequalities for the polar derivative of a polynomial with restricted zeros, Math. Bulk. 17 (2003), 15–28. Google Scholar

[5] A. Aziz and N. A. Rather, A refinement of a theorem of Paul Tur an concerning polynomials, Math. Ineq. Appl. 1 (1998), 231-238. Google Scholar

[6] C. Frappier, Q. I. Rahman and Rt. St. Ruscheweyh, New inequalities for poly- nomials, Trans, Amer. Math. Soc. 288 (1985),69–99. Google Scholar

[7] V. N. Dubinin, Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros, J. Math. Sci. 143 (2007), 3069–3076. Google Scholar

[8] N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543–546. Google Scholar

[9] P. D. Lax, Proof of a conjecture of P. Erd ̈os on the derivative of a polynomial, Bull. Amer. Math. Soc.,50(1944), 509–513. Google Scholar

[10] G. V. Milovanovi c, D. S. Mitrinovi c and Th. M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co., Singapore, (1994). Google Scholar

[11] R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128, 12 (2000), 3513–3517. Google Scholar

[12] Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York, 2002. Google Scholar

[13] N. A. Rather and Ishfaq Dar, Some applications of the Boundary Schwarz lemma for Polynomials with restricted zeros, Applied Mathematics E-Notes 20 (2020), 422–431. Google Scholar

[14] N. A. Rather, Ishfaq Dar and A. Iqbal, Some extensions of a theorem of Paul Tur an concerning polynomials, Kragujevac Journal of Mathematics, 46 (6), 969- 979. Google Scholar

[15] N. A. Rather, Ishfaq Dar and A. Iqbal, On a refinement of Tura n's inequality, Complex Anal Synerg. 6 (21) (2020). https://doi.org/10.1007/s40627-020-00058- 5. Google Scholar

[16] N. A. Rather and Suhail Gulzar, Generalization of an inequality involving maximum moduli of a polynomial and its polar derivative, Non linear Funct. Anal. and Appl. 19 (2014), 213–221. Google Scholar

[17] P. Tur an, Uber die Ableitung von Polynomen, Compos. Math. 7 (1939), 89-95. Google Scholar