arXiv subjects
Publications and source records attributed to S. H. Ahangar.
In this paper we obtain some refinements of a well-known result of Eneströ-Kakeya concerning the bounds for the moduli of the zeros of polynomials with complex coefficients which improve upon some results due to Aziz and Mohammad, Govil and Rahman and others.
Let $P(z)$ be a polynomial of degree $n$ having no zero in $|z|<k$ where $k\geq 1,$ then for every real or complex number $α$ with $|α|\geq 1$ it is known \begin{equation*} \underset{|z|=1}{\max}|D_αP(z)|\leq n\left(\dfrac{|α|+k}{1+k}\right)\underset{|z|=1}{\max}|P(z)|, \end{equation*} where $D_αP(z)=nP(z)+(α-z)P^{\prime}(z)$ denote the polar derivative of the polynomial $P(z)$ of degree $n$ with respect to a point $α\in\mathbb{C}.$ In this paper, by a simple method, a refinement of above inequality and other related results are obtained.