The extreme points of some classes of polynomials

Synopsis

We study the extreme points of two classes of polynomials of degree at most n:

It turns out that f ∈ Ext if and only if Re f(e^iθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈H_n and 1−|f(e^iθ)|² has 2n zeros in [0, 2π), then either f ∈ Ext H_n or else f(z) = α + βzⁿ where |α|+|β| = l and αβ≠0; if 1−|f(e^iθ)|² has 2m zeros, 2n, then f may or may not belong to Ext H_n.