Inequalities for the Polar Derivative of a Polynomial with Restricted Zeros

Ahmad Zireh, Mahmood Bidkham

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 Analysis and Operator Theory, 2010] that if $p(z)$ has no zeros in $|z|<1$ then for all $\alpha$, $\beta\in \mathbb{C}$ with $|\alpha|\geq 1$, $|\beta|\leq 1$ and $|z|=1$, \begin{align*} eftěrt zD_{lpha}p(z)+n\beta\frac{|lpha|-1}{2}p(z)\right\rverteq&\frac{n}{2}\Bigg\{\left[eftěrtlpha+\beta\frac{|lpha|-1}{2}\rightěrt+eftěrt z+\beta\frac{|lpha|-1}{2}\rightěrt\right] \max_{|z|=1}|p(z)| &\quad-eft[eftěrtlpha+\beta\frac{|lpha|-1}{2}\rightěrt-eftěrt z+\beta\frac{|lpha|-1}{2}\rightěrt\right] \min_{|z|=1}|p(z)|\Bigg\}. \end{align*} In this paper we extend above inequality for the polynomials having no zeros in $|z|<1$, except $s$-fold zeros at the origin. Our result generalize certain well-known polynomial inequalities.