Document Type : Original Article

Authors

Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India

Abstract

 Let a polynomial $P(z)$ of degree $n$ has all it's zeros in $|z|\leq 1.$  The Gauss-Lucas Theorem \cite{4}, asserts that all its  critical points also lie in $|z|\leq 1$ . Let $P(z^*)=0,$  then the famous Sendov's conjecture \cite{4}, says that the closed disk $|z-z^*|\leq 1$ contains a critical point of $P(z),$ (i.e. a zero of $P^\prime (z)$).  The conjecture has been proved for the polynomials of degree at most eight  \cite{2}. Also, the conjecture is true for some special classes of polynomials such as the polynomials having a zero at the origin and the polynomials having all their zeros on $|z|= 1$, as shown in  \cite{2}. However, the general version is still unproved.  A.Aziz\cite{1}, proved the following results regarding the relationship between the zeros and critical points of a polynomial.

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