On univalent polynomials with critical points on the unit circle

Proposition 1. If $n \ge 2$ and the polynomial $$Q(z) = 1 + c_1 z + c_2 z^2 + \dots + c_{n-1} z^{n-1} + z^n$$ has positive real part in $\mathbb{D}$ then $c_1 = c_2 = \ldots = c_{n-1} = 0$ .

Theorem 2. Let $P(z) = z + a_2 z^2 + ... + a_{n-1} z^{n-1} + a_n z^n$ be a polynomial in the class S with all critical points on the unit circle (that is, $|a_n| = \frac{1}{n}$ ). Then the following statements are equivalent: - (a) P' has positive real part in $\mathbb{D}$ . - (b) $a_2 = \ldots = a_{n-1} = 0$ . - (c) P is starlike. Note that P has any of the properties (a)–(c) if and only if any of its rotations $P_{\lambda}(z) = \overline{\lambda} P(\lambda z)$ , $|\lambda| = 1$ , has the corresponding property. Thus, without loss of generality we may assume that $P(z) = z + a_2 z^2 + ... + a_{n-1} z^{n-1} + \frac{z^n}{n}$ . We proceed under this assumption. We first show that (a) $\Leftrightarrow$ (b) and then also that (b) $\Leftrightarrow$ (c).