Generalization of certain well-known inequalities for the derivative of polynomials

For a polynomial P(z) of degree n which has no zeros in z < 1, Dewan et al. [4] established the inequality vertical bar zP'(z) + n beta/2P(z)vertical bar <= n/2 {(vertical bar beta/2 vertical bar + vertical bar 1+ beta/2 vertical bar) max(vertical bar z vertical bar = 1) P(z)vertical bar - (vertical bar 1 + beta/2 vertical bar - vertical bar beta/2 vertical bar) min(vertical bar z vertical bar=1) P(z)vertical bar} for any beta a parts per thousand currency sign 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 a parts per thousand currency sign 1. Our results generalize certain well-known polynomial inequalities.