The article introduces Ahlfors' generalization of Schwarz' lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings, including celebrated Picard's theorems. The article concludes with a brief insight into the theory of Kobayashi hyperbolic complex manifolds. Although this survey paper does not contain any new results, it may be useful for the beginner in complex analysis to better understands the concept of hyperbolicity in complex geometry.

Author Bio

Aleksander Simonic is a student of mathematics at the Faculty of Mathematics and Physics, University of Ljubljana. In his BSc thesis, he focused on hyperbolic complex geometry, which continues to attract his attention. The central aim of this article is to spark a nonexpert's interest in this exciting area of complex analysis. Aleksander is also concerned with the Riemann zeta function and the history of mathematics. He enjoys classical music, especially orchestral pieces (Mahler's symphonies).