Inequalities for the derivative of a polynomial

If p(z) = n ν=0 aνzν is a polynomial of degree n having all its zeros in |z| k, k 1, then Govil (Proc. Amer. Math. Soc. 41 (1973) 543–546) proved max |z|=1 |p (z)| n 1 +kn max |z|=1 |p(z)|. In this paper we generalize as well as improve upon the above inequality and a result due to Aziz (Proc. Amer. Math. Soc. 89 (1983) 259–266).  2002 Elsevier Science (USA). All rights reserved.