Some Sharpening and Generalizations of a result of T. J. Rivlin

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n$. Rivlin \cite{Rivlin} proved that if $p(z)\neq 0$ in the unit disk, then for $0<r\leq 1$, $\displaystyle{\max_{|z| = r}|p(z)|} \geq \Big(\dfrac{r+1}{2}\Big)^n \displaystyle{\max_{|z|=1} |p(z)|}.$ ~In this paper, we prove a sharpening and generalization of this result, and show by means of examples that for some polynomials our result can significantly improve the bound obtained by the Rivlin's Theorem.