# schlicht functions

###### Definition.

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 $f^{\prime}(0)=1$ is called the class of . 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

 $f(z):=\frac{g(z)-g(0)}{g^{\prime}(0)}$

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.

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.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title schlicht functions SchlichtFunctions 2013-03-22 14:23:37 2013-03-22 14:23:37 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 30C45 schlicht function KoebeDistortionTheorem Koebe14Theorem