A b s t r a c t . For a polynomial P (z) of degree n which has no zeros in |z| < 1, Dewan et al. [4] established the inequalityfor any |β| ≤ 1 and |z| = 1. In this paper we improve the above inequality for the sth derivative of a polynomial which has no zeros in |z| < k, k ≤ 1. Our results generalize certain well-known polynomial inequalities.