On Extremal Properties for the Polar Derivative of Polynomials

Abstract

If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k \leq 1$, then it is proved^[5] that $$\max_{|z|=1}|p′(z)| \leq\frac{n}{k^{n−1}+k^n}\max_{|z|=1}|p(z)|.$$In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_nz^n +\sum\limits_{j=\mu}^{n}c_{n-j}z^{n-j}$, $1 \leq \mu  \leq n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.