  Brannan showed that a normalized univalent polynomial of the form $P(z)=z+a_2
z^2+\ldots + a_{n-1}z^{n-1}+\frac{z^n}{n}$ is starlike if and only if
$a_2=\ldots=a_{n-1}=0$. 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 Fej\'er lemma for
trigonometric polynomials with positive real part.
