Univalent Functions: The Basics

This chapter provides a brief introduction to univalent function theory, describing the classes \(\mathbf {S}\) and \(\Sigma \) of univalent functions in the unit disk and in the complement of the closed disk, respectively. Having obtained the Area Theorem for the class \(\Sigma \) , Bieberbach’s estimate for the second coefficient \(a_2\) of functions in the class \(\mathbf {S}\) is proved and then the Koebe \(1/4\) -Theorem derived. As a consequence, the hyperbolic metric in a simply connected domain is shown to be comparable to the reciprocal of the distance to the boundary. The last section in the chapter covers the standard growth and distortion theorems for the class \(\mathbf {S}\) from which, in particular, we deduce that \(\mathbf {S}\) is compact.