Inequalities for the derivative of a polynomial

Abstract

Let P ( z ) = ∑ j = 0 n a j z j = c ∏ j = 1 n ( z − z j ) P(z) = \sum \nolimits _{j = 0}^n {{a_j}{z^j} = c\prod \nolimits _{j = 1}^n {(z - {z_j})} } be a polynomial of degree n n and P ′ ( z ) P’(z) its derivative. In this paper we consider the problem of estimating the maximum of | P ′ ( z ) | \left | {P’(z)} \right | on | z | = 1 \left | z \right | = 1 under some assumptions on the zeros or on the coefficients of P ( z ) P(z) and obtain certain sharp results.