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

 
 

 

Results on bi-univalent functions


Authors: D. Styer and D. J. Wright
Journal: Proc. Amer. Math. Soc. 82 (1981), 243-248
MSC: Primary 30C45; Secondary 30C75
DOI: https://doi.org/10.1090/S0002-9939-1981-0609659-5
MathSciNet review: 609659
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Abstract: When the class $ \sigma $ of bi-univalent functions was first defined, it was known that functions of the form $ \phi \circ {\psi ^{ - 1}} \in \sigma $ when $ \phi $ and $ \psi $ are univalent, map the unit disc $ {\mathbf{B}}$ onto a set containing $ {\mathbf{B}}$, and satisfy $ \phi (0) = \psi (0) = 0$, $ \phi '(0) = \psi '(0)$. It is shown here that such functions form a proper subset of $ \sigma $, and that $ \sigma $ is a proper subset of the set of functions of the form $ \phi \circ {\psi ^{ - 1}}$, where $ \phi $ and $ \psi $ are locally univalent, at most $ 2$-valent, each maps a subregion of $ {\mathbf{B}}$ univalently onto $ {\mathbf{B}}$, and $ \phi (0) = \psi (0) = 0$, $ \phi '(0) = \psi '(0)$, $ {\psi ^{ - 1}}(0) = 0$. It is also shown that there are $ f(z) = z + {a_2}{z^2} + \cdots $ in $ \sigma $ with $ \left\vert {{a_2}} \right\vert > 4/3$. However, doubt is cast that $ \left\vert {{a_2}} \right\vert$ can be as large as $ 3/2$.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1981-0609659-5
Article copyright: © Copyright 1981 American Mathematical Society

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