Inequalities for a polynomial and its derivative

In this paper we consider a class of polynomials P(z) = a(0) + (j=mu)Sigma(n) a(j)z(j), 1less than or equal tomuless than or equal ton, not vanishing in the disk \z\ < K. For K greater than or equal to 1, we investigate the dependence of max(\z\=1) \P(Rz)- P(z)\ on max(\z\=1) \P(z)\ and for K > 0 we estimate max(\z\=R)\P'(z)\ in terms of max(\z\=r) \P(z)\, 0 less than or equal to r less than or equal to R less than or equal to K. Our results not only generalize some known polynomial inequalities, but olso a variety of interesting results can be deduced from these by a fairly uniform procedure. We also obtain a generalizati on of a Theorem of Paul Turan.