Geometric function theory is the study of geometric

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 properties of analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s. A fundamental result in the theory is the Riemann mapping theorem.

Riemann mapping theorem

Let z be a point in a simply-connected region D (D≠ ℂ) and D having at least two boundary points. Then there exists a unique analytic function w = f(z) mapping D bijectively into the open unit disk |w|<1 such that f(z)=0 and
f ′(z)>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 and D as two simply connected regions different from ℂ. The Riemann mapping theorem provides the existence of w=f(z) mapping D onto the unit disk and existence of w=g(z) mapping D onto the unit disk. Thus gf is a one-one mapping of D onto D.
If we can show that g, and consequently the composition, is analytic, we then have a conformal mapping of D onto D, proving "any two simply connected regions different from the whole plane ℂ can be mapped conformally onto each other."

Univalent function

We know that a complex function is a multiple valued function. That is, for distinct points z, z,... in a domain D, they may share a common value, f(z)=f(z)=... But if we restrict a complex function to be injective( one-one ), then we obtain a class of functions, viz, univalent functions. 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 and z in D, i.e f(z)≠f(z) implies z≠z. Alternate terms in common use are schilicht and simple. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.