# Loewner chain

A Loewner chain is a continuous function $f:{\mathbb{D}}\times[0,\infty)\rightarrow{\mathbb{C}}$ (where ${\mathbb{D}}$ refers to the open unit disc) with the following properties:

1. 1.

the function $z\mapsto f(z,t)$ is analytic and univalent for all $t\in[0,\infty)$

2. 2.

$f(0,t)=0$ and $f^{\prime}(0,t)=e^{t}$

3. 3.

$f({\mathbb{D}},s)\subseteq f({\mathbb{D}},t)$ for $0\leq s

Bibliography

• John B. Conway . Springer-Verlag, New York, Inc, 1995

Title Loewner chain LoewnerChain 2013-03-22 14:12:02 2013-03-22 14:12:02 jirka (4157) jirka (4157) 4 jirka (4157) Definition msc 30C55