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schlicht functions
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(Definition)
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Definition 1 The class of univalent functions on the open unit disc in the complex plane such that for any $f$ in the class we have $f(0) = 0$ and $f'(0) = 1$ is called the class of schlicht functions. Usually this class is denoted by ${\mathcal{S}}$ .
Note that if $g$ is any univalent function on the unit disc, then the function $f$ defined by \begin{equation*} f(z): = \frac{g(z) - g(0)}{g'(0)} \end{equation*}belongs to ${\mathcal{S}}$ . So to study univalent functions on the unit disc it suffices to study ${\mathcal{S}}$ . A basic result on these gives that this set is in fact compact in the space of analytic functions on the unit disc.
Theorem 1 Let $\{ f_n \}$ be a sequence of functions in ${\mathcal{S}}$ and $f_n \to f$ uniformly on compact subsets of the open unit disc. Then $f$ is in ${\mathcal{S}}$ .
Alternatively this theorem can be stated for all univalent functions by the above remark, but there a sequence of univalent functions can converge either to a univalent function or to a constant. The requirement that the first derivative is 1 for functions in ${\mathcal{S}}$ prevents this problem.
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- John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.
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"schlicht functions" is owned by jirka.
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Cross-references: first derivative, converge, theorem, compact subsets, sequence, space of analytic functions, compact, belongs, function, complex plane, unit disc, open, univalent functions, class
There are 4 references to this entry.
This is version 5 of schlicht functions, born on 2004-06-04, modified 2005-03-07.
Object id is 5890, canonical name is SchlichtFunctions.
Accessed 3529 times total.
Classification:
| AMS MSC: | 30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions ) |
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Pending Errata and Addenda
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