univalent analytic function

Definition.

An analytic function on an open set is called univalent if it is one-to-one.

For example mappings of the unit disc to itself $\\phi_{a}:{\\mathbb{D}}\\rightarrow{\\mathbb{D}}$ , where $\\phi_{a}(z)=\\frac{z-a}{1-\\bar{a}z}$ , for any $a\\in{\\mathbb{D}}$ are univalent.The following summarizes somebasic of univalent functions.

Proposition.

Suppose that $G,\\Omega\\subset{\\mathbb{C}}$ are regions and $f\\colon G\\to\\Omega$ is a univalent mapping such that $f(G)=\\Omega$ (itis onto), then

•
$f^{-1}\\colon\\Omega\\to G$ (where $f^{-1}(f(z))=z$ ) is an analyticfunction and $(f^{-1})^{\\prime}(f(z))=\\frac{1}{f^{\\prime}(z)}$ ,
•
$f^{\\prime}(z)\ot=0$ for all $z\\in G$

References

1 John B. Conway..Springer-Verlag, New York, New York, 1978.
2 John B. Conway..Springer-Verlag, New York, New York, 1995.

单词	UnivalentAnalyticFunction
释义	univalent analytic function Definition. An analytic function on an open set is called univalent if it is one-to-one. For example mappings of the unit disc to itself $\\phi_{a}:{\\mathbb{D}}\\rightarrow{\\mathbb{D}}$ , where $\\phi_{a}(z)=\\frac{z-a}{1-\\bar{a}z}$ , for any $a\\in{\\mathbb{D}}$ are univalent.The following summarizes somebasic of univalent functions. Proposition. Suppose that $G,\\Omega\\subset{\\mathbb{C}}$ are regions and $f\\colon G\\to\\Omega$ is a univalent mapping such that $f(G)=\\Omega$ (itis onto), then • $f^{-1}\\colon\\Omega\\to G$ (where $f^{-1}(f(z))=z$ ) is an analyticfunction and $(f^{-1})^{\\prime}(f(z))=\\frac{1}{f^{\\prime}(z)}$ , • $f^{\\prime}(z)\ot=0$ for all $z\\in G$ References 1 John B. Conway..Springer-Verlag, New York, New York, 1978. 2 John B. Conway..Springer-Verlag, New York, New York, 1995.
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