# Geometric function theory

Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

## Riemann mapping theorem

Let $z_0$ be a point in a simply-connected region $D_1 (D_1 \neq \mathbb{C})$ and $D_1$ having at least two boundary points. Then there exists a unique analytic function $w=f(z)$ mapping $D_1$ bijectively into the open unit disk $|w| < 1$ such that $f(z_0)=0$ and $f'(z_0) > 0$.

It should be noted that while Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function.

### Elaboration

In the above figure, consider $D_1$ and $D_2$ as two simply connected regions different from $\mathbb C$. The Riemann mapping theorem provides the existence of $w=f(z)$ mapping $D_1$ onto the unit disk and existence of $w=g(z)$ mapping $D_2$ onto the unit disk. Thus $g^{-1}f$ is a one-to-one mapping of $D_1$ onto $D_2$. If we can show that $g^{-1}$, and consequently the composition, is analytic, we then have a conformal mapping of $D_1$ onto $D_2$, proving "any two simply connected regions different from the whole plane $\mathbb C$ can be mapped conformally onto each other."

## Univalent function

Of special interest are those complex functions which are one-to-one. That is, for points $z_1$, $z_2$, in a domain $D$, they share a common value, $f(z_1)=f(z_2)$ only if they are the same point $z_1=z_2$. A function $f$ analytic in a domain $D$ is said to be univalent there if it does not take the same value twice for all pairs of distinct points $z_1$ and $z_2$ in $D$, i.e. $f(z_1) \neq f(z_2)$ implies $z_1 \neq z_2$. Alternate terms in common use are schlicht( this is German for plain, simple) and simple. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.

## References

• Krantz, Steven (2006). Geometric Function Theory: Explorations in Complex Analysis. Springer. ISBN 0-8176-4339-7.
• Noor, K.I. Lecture notes on Introduction to Univalent Functions. CIIT, Islamabad, Pakistan.
• Bulboacă, T.; Cho, N. E.; Kanas, S. A. R. (2012). "New Trends in Geometric Function Theory 2011". International Journal of Mathematics and Mathematical Sciences 2012: 1. doi:10.1155/2012/976374. edit
• Ahlfors, Lars (2010). Conformal Invariants: Topics in Geometric Function Theory. AMS Chelsea Publishing. ISBN 978-0821852705. edit