Some Integral Mean Estimates for Polynomials

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New Zealand Journal of Mathematics

Vol. 44, (2014), Pages 83-91


N. A. Rather

Department of Mathematics,

University of Kashmir,

Harzarbal, Sringar 190006, India.

mailto:dr.narather@gmail.com

Suhail Gulzar

Department of Mathematics,

University of Kashmir,

Harzarbal, Sringar 190006, India.


mailto:sgmattoo@gmail.com

K. A. Thakur

Department of Mathematics,

University of Kashmir,

Harzarbal, Sringar 190006, India.


mailto:thakurkhursheed@gmail.com


Abstract For the class of Lacunary polynomials P(z)=a_nz^n+\sum_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu}, 1\leq\mu\leq n, of degree n having all their zeros in |z|\leq k where k\leq 1, Aziz and Shah proved for each r > 0

n\Bigg\{\int\limits_{0}^{2\pi}\left|P\left(e^{i\theta}\right)\right|^{r} d\theta\Bigg\}^{\frac{1}{r}}\leq\Bigg\{ \int\limits_{0}^{2\pi}\left|1+k^{\mu}e^{i\theta}\right|^{r} d\theta\Bigg\}^{\frac{1}{r}}{\max}_{|z|=1}|P^{\prime}(z)|.


In this paper, we extend above inequality to the polar derivative thereby establish some refinements and generalizations of some known polynomial inequalities concerning the polar derivative of a polynomial with restricted zeros.

Keywords Polynomials; Inequalities in the complex domain; Polar derivative.

Classification (MSC2000) 30C10, 30A10, 41A17.

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