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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc


Author: Stephan Ruscheweyh
Journal: Trans. Amer. Math. Soc. 210 (1975), 63-74
MSC: Primary 30A40; Secondary 30A10
MathSciNet review: 0382626
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Abstract: Let $ A$ be the set of functions regular in the unit disc $ \mathcal{U}$ and $ {A_0}$ the set of all functions $ f \in A$ which satisfy $ f(0) = 1$. For $ V \subset {A_0}$ define the dual set $ {V^ \ast } = \{ f \in {A_0}\vert f \ast g \ne 0{\text{ for all }}g \in V,z \in \mathcal{U}\} ,{V^{ \ast \ast }} = {({V^ \ast })^ \ast }$. Here $ f \ast g$ denotes the Hadamard product. THEOREM. Let $ V \subset {A_0}$ have the following properties: (i) $ V$ is compact, (ii) $ f \in V$ implies $ f(xz) \in V$ for all $ \vert x\vert \leqslant 1$. Then $ \lambda (V) = \lambda ({V^{ \ast \ast }})$ for all continuous linear functionals $ \lambda $ on $ A$. This theorem has many applications to functions in $ A$ which are defined by properties like bounded real part, close-to-convexity, univalence etc.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0382626-7
PII: S 0002-9947(1975)0382626-7
Keywords: Hadamard product, continuous linear functionals, functions with bounded real part, close-to-complex functions, partial sums, Marx conjecture
Article copyright: © Copyright 1975 American Mathematical Society