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Results on BI-univalent Functions

D. Styer and D. J. Wright
Proceedings of the American Mathematical Society
Vol. 82, No. 2 (Jun., 1981), pp. 243-248
DOI: 10.2307/2043317
Stable URL: http://www.jstor.org/stable/2043317
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Results on BI-univalent Functions
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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 $f(z) = z + a_2z^2 + \cdots$ in $\sigma$ with $|a_2| > 4/3$. However, doubt is cast that $|a_2|$ can be as large as 3/2.

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