Operators and growth of polynomials

Abstract: For a polynomial \(P(z)\)=\[\sum_{v=0}^{n} a{_v}z{^u}\], of degree n. A famous result known as Bernstein's inequality states that if \(P(z)\) is a polynomial of degree n; then  \[ \max_{|z|=1}\] \(|P'(z)|\)  \(n\) \[ \max_{|z|=1}\]  \(|P(z)|\). 

The result is best possible and attains equality for polynomials of the form \(P(z)\) = α \[z{^n}\], α 0. In this talk we will present the refinements and generalizations of the above inequality and also try to find the relationship between the bound n and the distance of the zeros of the polynomial from the origin. Finally we present the \(lr\) analogue of the above inequality and discuss the polar derivative of the polynomial \(P(z)\).