On Zero Free Regions for Derivatives of a Polynomial

Mohammad Ibrahim Mir, Ishfaq Nazir, Irfan Ahmad Wani

Let $P_{n}$ denote the set of polynomials of the form $$p(z)= (z-a)^{m} rod_{k=1}^{n-m} (z-z_{k}),$$ with $|a|\leq 1$ and $|z_{k}| \geq 1$ for $1\leq k \leq n-m.$ For the polynomials of the form $p(z)= z \prod_{k=1}^{n-1} (z-z_{k}), $ with $|z_{k}| \geq 1$, where $1\leq k \leq n-1$, Brown \cite{AO1} stated the problem ``Find the best constant $C_{n}$ such that $p'(z)$ does not vanish in $|z|< C_{n}$''. He also conjectured in the same paper that $C_{n} = \frac{1}{n}$. This problem was solved by Aziz and Zarger \cite{Alt}. In this paper, we obtain the results which generalizes the results of Aziz and Zarger.