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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Injective matrix functions


Author: Binyamin Schwarz
Journal: Proc. Amer. Math. Soc. 83 (1981), 331-336
MSC: Primary 30C55; Secondary 15A54, 30G30
DOI: https://doi.org/10.1090/S0002-9939-1981-0624924-3
MathSciNet review: 624924
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Abstract: Univalence of holomorphic (scalar) functions $ f(z)$ is generalized to injectivity of holomorphic matrix functions $ V(z) = ({\upsilon _{ik}}(z))_1^n$. Local injectivity is characterized by $ \left\vert {V'({z_0})} \right\vert \ne 0\left( {\left\vert A \right\vert = \det A} \right)$. The classes $ S$ and $ \Sigma $ are defined as in the scalar case. For each class a sufficient condition is proved and a necessary condition is conjectured.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0624924-3
Keywords: Univalent functions, injective matrix functions
Article copyright: © Copyright 1981 American Mathematical Society