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schlicht functions (Definition)
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

$\displaystyle f(z): = \frac{g(z) - g(0)}{g'(0)}$    

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.

Bibliography

1
John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.



"schlicht functions" is owned by jirka.
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See Also: Koebe distortion theorem, Koebe 1/4 theorem

Other names:  schlicht function
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Cross-references: first derivative, converge, compact subsets, sequence, space of analytic functions, compact, 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 2973 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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