On Erdös-lax inequality concerning polynomials

Recently Milovanovic et al. [Bulletin T.CLIII de l?Aca?mie 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 Milovanovic.