TY - JOUR
T1 - On Extremal Properties for the Polar Derivative of Polynomials
JO - Analysis in Theory and Applications
VL - 2
SP - 150
EP - 157
PY - 2011
DA - 2011/04
SN - 27
DO - http://doi.org/10.1007/s10496-011-0150-3
UR - https://global-sci.org/intro/article_detail/ata/4588.html
KW - polynomial, zeros, inequality, polar derivative.
AB - <p style="text-align: justify;">If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k \leq 1$, then it is proved<sup>[5]</sup> 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 $p(z) = c_nz^n +\sum\limits_{j=\mu}^{n}c_{n-j}z^{n-j}$, $1 \leq \mu &nbsp;\leq n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.</p>