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

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 ${\displaystyle D_{1}}$  and ${\displaystyle D_{2}}$  as two simply connected regions different from ${\displaystyle \mathbb {C} }$ . The Riemann mapping theorem provides the existence of ${\displaystyle w=f(z)}$  mapping ${\displaystyle D_{1}}$  onto the unit disk and existence of ${\displaystyle w=g(z)}$  mapping ${\displaystyle D_{2}}$  onto the unit disk. Thus ${\displaystyle g^{-1}f}$  is a one-to-one mapping of ${\displaystyle D_{1}}$  onto ${\displaystyle D_{2}}$ . If we can show that ${\displaystyle g^{-1}}$ , and consequently the composition, is analytic, we then have a conformal mapping of ${\displaystyle D_{1}}$  onto ${\displaystyle D_{2}}$ , proving "any two simply connected regions different from the whole plane ${\displaystyle \mathbb {C} }$  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 ${\displaystyle z_{1}}$ , ${\displaystyle z_{2}}$ , in a ___domain ${\displaystyle D}$ , they share a common value, ${\displaystyle f(z_{1})=f(z_{2})}$  only if they are the same point ${\displaystyle z_{1}=z_{2}}$ . A function ${\displaystyle f}$  analytic in a ___domain ${\displaystyle D}$  is said to be univalent there if it does not take the same value twice for all pairs of distinct points ${\displaystyle z_{1}}$  and ${\displaystyle z_{2}}$  in ${\displaystyle D}$ , i.e ${\displaystyle f(z_{1})\neq f(z_{2})}$  implies ${\displaystyle z_{1}\neq z_{2}}$ . 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.