Distribution of Zeros and Critical Points of a Polynomial, and Sendov's Conjecture

According to the Gauss-Lucas theorem, the critical points of a complex polynomial p(z):= Sigma(n )(j=0)a(j)z(j) where a(j) always lie in the convex hull of its zeros. In this paper, we prove certain relations between the distribution of zeros of a polynomial and its critical points. Using these relations, we prove the well-known Sendov's conjecture for certain special cases.