Chapter 4 - Bernstein-type inequalities for polynomials with restricted zeros

Abstract

The Bernstein inequality states that if p(z) is a polynomial of degree at most n, then ‖p′‖⩽n‖p‖ where ‖p‖=max|z|=1⁡|p(z)|. This inequality is sharp and equality holds if and only if p(z)=λzn for some complex λ. So if p(z) is a polynomial that does not have all its zeros at z=0, then Bernstein's inequality can be improved. For example, Erdős conjectured and Lax proved that for p(z) a polynomial of degree at most n such that p(z)≠0 for |z|<1, we have ‖p′‖⩽n‖p‖/2. This chapter considers Bernstein-type inequalities for polynomials which have restrictions on the location of their zeros (often restricting the zeros to lie inside or outside of a disk). It also considers higher-order derivatives, refinements involving min|z|=K⁡|p(z)|, and results incorporating the moduli of coefficients and/or zeros of the polynomial. Besides, inequalities for self-inversive and self-reciprocal polynomials, as well as for some other classes of polynomials, are investigated.