like a sphere or a torus or several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Extremal problems Topics in this area include "Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations". Univalent and multivalent functions A holomorphic function on an open subset of the complex plane is called univalent if it is injective. One can prove that if and are two open connected sets in the complex plane, and is a univalent function such that (that is, is surjective), then the derivative of is never zero, is invertible, and its inverse is also holomorphic. More, one has by the chain rule 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. Important theorems Riemann mapping theorem Let be a point in a simply-connected region and having at least two boundary points. Then there exists a unique analytic function mapping bijectively into the open unit disk such that and . Although Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function. An example is given below. In the above figure, consider and as two simply connected regions different from . The Riemann mapping theorem provides the existence of mapping onto the unit disk and existence of mapping onto the unit disk. Thus is a one-to-one mapping of onto . If we can show that , and consequently the composition, is analytic, we then have a conformal mapping of onto , proving "any two simply connected regions different from the whole plane can be mapped conformally onto each other." Schwarz's Lemma The Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem, which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic functions. Statement Schwarz Lemma. Let D = {z : |z| < 1} be the open unit disk in the complex plane