# 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 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

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

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 $\mathrm{\{}{f}_{n}\mathrm{\}}$ be a sequence of functions in $\mathrm{S}$ and ${f}_{n}\mathrm{\to}f$ uniformly on compact subsets of the open unit disc. Then $f$ is in $\mathrm{S}$.

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 |
---|---|

Canonical name | SchlichtFunctions |

Date of creation | 2013-03-22 14:23:37 |

Last modified on | 2013-03-22 14:23:37 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30C45 |

Synonym | schlicht function |

Related topic | KoebeDistortionTheorem |

Related topic | Koebe14Theorem |