On Extremal Properties for the Polar Derivative of Polynomials (Analysis in Theory and Applications, 2011)
If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k ≤ 1$, then it is proved[5] that $$\max_{|z|=1}|p′(z)| ≤\frac{n}{k^{n−1}+k^n}\max_{|z|=1}|p(z)|.$$In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_nz^n +\sum\limits_{j=μ}^{n}c_{n-j}z^{n-j}$, $1 ≤ μ ≤ n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.