Univalent functions and area problems

Abstract

This thesis contains a survey of basic properties of univalent functions in the analytic function theory. Mostly we focuses on the class of univalent functions in the unit disk in which each of them has a Taylor’s series expansion with a specific normalized form. This class of functions is preserved under certain elementary transformations. The well-known Bieberbach theorem, the growth theorem, the distortion theorem, the Koebe 1/4-theorem, area theorems are presented in this thesis. The classical subclasses of univalent functions, namely, the class of convex and starlike functions are also studied including their characterizations. As a part of applications of above and other related properties considered in this thesis, we compute areas of image domains of the unit disk and its subdisks under functions of some special types looking into the fact that the image domains are bounded. These are also examined through several examples of functions and their graphs. Finally, in the line of area of regions, we expect that a number of problems can be studied to maximize length of image of unit circle over the class of univalent functions. A few analysis on the latter part are covered in the concluding chapter.