On univalent polynomials with critical points on the unit circle

Abstract

Brannan showed that a normalized univalent polynomial of the form (P(z)=z+a2 z2+...+ an-1zn-1+znn) is starlike if and only if (a2=...=an-1=0). We give a new and simple proof of his result, showing further that it is also equivalent to the membership of P in the Noshiro–Warschawski class of univalent functions whose derivative has positive real part in the disk. Both proofs are based on the Fejér lemma for trigonometric polynomials with positive real part.