Inequalities for the Polar Derivative of a Polynomial

Abstract

For a polynomial $p\left(z\right)$ of degree $n$, we consider an operator $D_{\alpha }$ which map a polynomial $p\left(z\right)$ into $D_{\alpha }p\left(z\right):=(\alpha -z)p\mathrm{\text{'}}\left(z\right)+np\left(z\right)$ with respect to $\alpha$. It was proved by Liman et al. (2010) that if $p\left(z\right)$ has no zeros in then for all $\alpha ,\mathrm{ }\mathrm{ }\beta \in \mathbb{C}$ with $\left|\alpha \right|\ge 1,\mathrm{ }\mathrm{ }\left|\beta \right|\le 1$ and $\left|z\right|=1$, $\left|z{D}_{\alpha }p\right(z)+n\beta (\left(\right|\alpha |-1)/2\left)p\right(z\left)\right|\le (n/2)\left\{\right[|\alpha +\beta (\left(\right|\alpha |-1)/2\left)\right|+|z+\beta (\left(\right|\alpha |-1)/2\left)\right|\left]{\text{m}\text{a}\text{x}}_{\left|z\right|=1}\right|p\left(z\right)|-[|\alpha +\beta (\left(\right|\alpha |-1)/2\left)\right|-|z+\beta (\left(\right|\alpha |-1)/2\left)\right|\left]{\text{m}\text{i}\text{n}}_{\left|z\right|=1}\right|p\left(z\right)\left|\right\}$. In this paper we extend the above inequality for the polynomials having no zeros in , where $k\le 1$. Our result generalizes certain well-known polynomial inequalities.