Some inequalities concerning the rate of growth of polynomials

In this paper we consider a class of polynomials p(z)=a_0+suml_{v=t}^n a_vz^v, tge1, not vanishing in |z| k, k ≥ 1 and investigate the dependence of max |z|=1 |p(Rz) − p(z)| on max |z|=1 |p(z)| . Our result not only generalizes some polynomial inequalities, but also a variety of interesting results can be deduced from it by a fairly uniform procedure.