Some inequalities for sth derivative of polynomials

Let P(z) be a polynomial of degree n which have no zeros in vertical bar z vertical bar<k,k >= 1. Then it was proved by Govil (J Approx Theory 66:29-35, 1991), for 1 <= s<n, then max(|z|=1)|P-(s)(z)|<= n(n-1) center dot center dot center dot/1+k(s)(max(|z|=1)|P(z)|-min(|z|=k|)P(z)|). In this paper, we shall present an generalization of this result and an L-gamma analogue of our result, which improve some previous results.