Distortion theorems for polynomials on a circle

Inequalities for the derivatives with respect to $\varphi=\arg z$ the functions $\operatorname{Re}P(z)$, $|P(z)|^2$ and $\arg P(z)$ are established for an algebraic polynomial $P(z)$ at points on the circle $|z|=1$. These estimates depend, in particular, on the constant term and the leading coefficient of the polynomial $P(z)$ and improve the classical Bernstein and Turan inequalities. The method of proof is based on the techniques of generalized reduced moduli.