# Some Integral Mean Estimates for Polynomials

### New Zealand Journal of Mathematics

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

N. A. Rather

Department of Mathematics,

University of Kashmir,

Harzarbal, Sringar 190006, India.

Suhail Gulzar

Department of Mathematics,

University of Kashmir,

Harzarbal, Sringar 190006, India.

K. A. Thakur

Department of Mathematics,

University of Kashmir,

Harzarbal, Sringar 190006, India.

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.