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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: A. Horwitz and D. J. Newman
Journal: Trans. Amer. Math. Soc. 295 (1986), 699-713
MSC: Primary 30C75; Secondary 30D50
MathSciNet review: 833704
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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 $ \Vert g\prime\Vert _{\infty } \leq 1$. Then we prove that $ \vert g(z)\vert \leq \vert\phi (z)\vert$ 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 $.

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Article copyright: © Copyright 1986 American Mathematical Society