A note on the zeros of polar derivative of a polynomial

Abstract

In $[4,7]$, Enestrom Kakeya theorem has stated as the following. If $f(z)=\sum_{j=0}^n k_j z^j$ is the $n^{t h}$ degree polynomial with real coefficients such that $0<k_0 \leq k_1 \leq \ldots \leq k_{n-2} \leq k_{n-1} \leq k_n$ then all zeros of $\mathrm{f}(\mathrm{z})$ lies in $|z| \leq 1$. In [1], Aziz and Mahammad, showed that zeros of $f(z)$ satisfies $|z| \geq \frac{n}{n+1}$ are simple, under the same conditions. In this paper, we extend the above result to the polar derivative by relaxing the hypothesis in different ways.