On the Derivative of a Polynomial with Prescribed Zeros

Abstract

For a polynomial p(z) = an ∏n t=1(z − zt) of degree n having all its zeros in |z| ≤ K, K ≥ 1 it is known that max |z|=1 |p′(z)| ≥ 2 1 + Kn { n ∑ t=1 K K + |zt| } max |z|=1 |p(z)| . By assuming a possible zero of order m, 0 ≤ m ≤ n− 4, at z = 0, of p(z) for n ≥ k + m + 1 with integer k ≥ 3 we have obtained a new refinement of the known result.