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