An extremal problem for analytic functions with prescribed zeros and $r$th derivative in $H^ \infty$

Abstract:

Let $({\alpha _1}, \ldots ,{\alpha _n})$ be $n$ points in the unit disc $U$. Suppose $g$ is analytic in $U$, $g({\alpha _1}) = \cdots = g({\alpha _n}) = 0$ (multiplicities included), and $\|g\prime \|_{\infty } \leq 1$. Then we prove that $|g(z)| \leq |\phi (z)|$ for all $z \in U$, where $\phi ({\alpha _1}) = \cdots = \phi ({\alpha _n}) = 0$ and $\phi \prime (z)$ is a Blaschke product of order $n - 1$. We extend this result in a natural way to convex domains $D$ with analytic boundary. For $D$ not convex we show that there is no extremal function $\phi$.