Recently Milovanović et al. [Bulletin T.CLIII de l'Académie Serbe des sciences et des arts - 2020.] proved that if P(z) ∈ Pn with no zeros in |z| < k, k ≥ 1, then,∣∣∣P′(z)∣∣∣ ≤ ∥P∥ 2 [ n − { n ( k − 1 k + 1 ) + 2 k + 1 ( |c0| − kn|cn| |c0| + kn|cn| )} |P(z)|2 ∥P∥2 ] , |z| = 1, where P(z) = c0 + c1z + · · · + cnzn ∈ Pn is a polynomial of degree n. In this paper, we obtain some results concerning the class of polynomials having s−fold zero at origin. These results not only generalizes but also refines many well-known results due to Milovanović.