Let $B$ be a finite Blaschke product of degree $n$. Suppose there exist an open subset $U$ of $\mathbb{D}$ such that $B$ is univalent on $U.$ Is it true that $n=1$ ?
2026-02-23 10:45:31.1771843531
Univalent Blaschke Products on $\mathbb{D}$
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No. As a holomorphic function, $B$ is univalent in some neighborhood of any point $z_0$ with $B'(z_0) \ne 0$, and for a non-constant function the zeros of the derivative are isolated.