# On univalent polynomials with critical points on the unit circle

## TL;DR

This paper provides a new, simplified proof that certain normalized univalent polynomials with a specific form are starlike if and only if their coefficients vanish, linking this to the Noshiro-Warschawski class.

## Contribution

The paper introduces a new, straightforward proof of Brannan's characterization of univalent polynomials with critical points on the unit circle, connecting it to the Noshiro-Warschawski class.

## Key findings

- Proof based on Fejér's lemma for trigonometric polynomials
- Equivalence between starlikeness and coefficients vanishing
- Connection to the Noshiro-Warschawski class

## Abstract

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.