Maximum Modulus of Polynomials

Abstract

Let $$P(z)= \sum_{j=0}^{n}a_j z^j$$ be a polynomial of degree $n$ and let $M(P,r)=\underset{|z|=r}{\max} |P(z)|$. If $P(z)\neq 0$ in $|z|<1$, then $$M(P,r)\geq {\bigg(\frac{1+r}{1+\rho}\bigg)^n}M(P,\rho).$$The result is best possible. In this paper we shall present a refinement of this result and some other related results.