(untitled) (Analysis in Theory and Applications, 2016)
Let p(z) be a polynomial of degree n, which has no zeros in |z|< 1, Dewan et al. [K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 (2010), pp. 38-41] established |zp'(z)+nβ2p(z)| ≤ n2{(|β2|+|1+β2|)_|z|=1|p(z)|-(|1+β2|-|β2|)_|z|=1|p(z)|}, for any |β|≤ 1 and |z|=1. In this paper we improve the above inequality for the polynomial which has no zeros in |z|< k, k≥ 1, except s-fold zeros at the origin. Our results generalize certain well known polynomial inequalities.