Some Inequalities for the Polar Derivative of a Polynomial Having S-Fold Zeros at the Origin

Document Type : Original Article


Department of Mathematics, University of Kashmir, Srinagar 190006, Jammu and Kashmir, India


Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|\leq 1$ then for all $(\alpha_i)^t_{i=1}\in \mathbb{C}$ with $|\alpha_i|\geq 1, 1\leq i\leq t<n$, it was proved by Jain[V. K. Jain, Generalization of an inequality involving maximum moduli of a polynomial and its polar derivative, Bull Math Soc Sci Math Roum Tome. 98, 67–74 (2007)] that
\max\limits_{|z|=1}|D_{\alpha_t}...D_{\alpha_2}D_{\alpha_1}P(z)|\geq\frac{n_t}{2^t}\left[A_{\alpha_t}\max\limits_{|z|=1}|P(z)|+\left( 2^t\prod\limits_{i=1}^{t}|\alpha_i|- A_{\alpha_t}\right)\min\limits_{|z|=1}|P(z)| \right].
where $n_t=n(n-1)...(n-t+1)$ and $A_{\alpha_t}=(|\alpha_1|-1)(|\alpha_2|-1)...(|\alpha_t|-1)$. In this paper, we generalize this and some other results.