On the location of zeros of generalized derivative

Abstract

Let $P(z) =displaystyle prod_{v=1}^n (z-z_v),$ be a monic polynomial of degree $n$, then, $G_gamma[P(z)] = displaystyle sum_{k=1}^n gamma_k prod_{{v=1},{v neq k}}^n (z-z_v),$ where $gamma= (gamma_1,gamma_2,dots,gamma_n)$ is a n-tuple of positive real numbers with $sum_{k=1}^n gamma_k = n$, be its generalized derivative. The classical Gauss-Lucas Theorem on the location of critical points have been extended to the class of generalized derivativecite{g}. In this paper, we extend the Specht Theorem and the results proved by A.Aziz cite{1} on the location of critical points to the class of generalized derivative .