Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings.
Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem.
There are several connections to mathematical physics, because of the relations to potential theory (in the plane). The Handbook of Geometric Function Theory contains also an article about constructive methods and further a Bibliography including applications eg: to electroxtatic problems, heat conduction, potential flows (in the plane).
- A collection of independent survey articles in the field of GeometricFunction Theory
- Existence theorems and qualitative properties of conformal and quasiconformal mappings
- A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane).

Geometric Function Theory is a central part of Complex Analysis (one complex variable). The Handbook of Complex Analysis - Geometric Function Theory deals with this field and its many ramifications and relations to other areas of mathematics and physics. The theory of conformal and quasiconformal mappings plays a central role in this Handbook, for example a priori-estimates for these mappings which arise from solving extremal problems, and constructive methods are considered. As a new field the theory of circle packings which goes back to P. Koebe is included. The Handbook should be useful for experts as well as for mathematicians working in other areas, as well as for physicists and engineers.
- A collection of independent survey articles in the field of GeometricFunction Theory
- Existence theorems and qualitative properties of conformal and quasiconformal mappings
- A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane)