Lq mean extension for the polar derivative of a polynomial

For a polynomial p(z)  of degree n, we consider an operator D_{\alpha} which map a polynomial p(z)  into D_{\alpha}p(z):=(\alpha-z)p'(z)+np(z) with respect to \alpha. It was proved by Liman et al [ A. Liman, R. N. Mohapatra and W. M. Shah, Inequalities for the polar derivative of a polynomial, Complex Anal. Oper. Theory, ۲۰۱۲] that if p(z) has no zeros in |z|<۱ then  for all \alpha,\ \beta\in \mathbb{C} with |\alpha|\geq ۱ , \ |\beta|\leq ۱ and |z|=۱,\begin{align*}\begin{split}|zD_{\alpha}p(z)+n\beta\frac{|\alpha|-۱}{۲}&p(z)|\leq \frac{n}{۲}\{[|\alpha+\beta\frac{|\alpha|-۱}{۲}|+|z+\beta\frac{|\alpha|-۱}{۲}|]\max_{|z|=۱}|p(z)|. \end{split}\end{align*} In this paper, we present the integral L_q mean extension of the above inequality for the polar derivative of polynomials. Our result generalize certain well-known polynomial inequalities.

Keywords:

Polynomial , Integral inequality , Polar derivative , Restricted Zeros