On the zeros and critical points of a polynomial

Let P(z)=a_۰ + a_۱z + \dots  + a_{n-۱}z^{n-۱}+z^n be a polynomial of degree n.  The Gauss-Lucas Theorem asserts that the zeros of the derivative P^\prime (z)= a_۱ + \dots  +(n-۱) a_{n-۱}z^{n-۲}+nz^{n-۱},  lie in the convex hull of the zeros of   P(z). Given a zero of  P(z) or P^\prime (z),  A. Aziz [۱], determined regions which contain at least one zero of  P(z) or P^\prime (z) respectively. In this paper, we give simple proofs and improved version of various results proved in [۱], concerning the zeros of a polynomial and its derivative.

Keywords:

Polynomial , Zeros , critical points , half plane , circular region