The main results on sufficient cjnditions for an analytic function to be schlicht

This article gives an account of the results on sufficient conditions for schlichtness of analytic functions that have been obtained from the time (in the thirties) when this line of investigation started in the geometric theory of functions, up to the present. A brief historical survey is given at the end.
In the first and second chapters we describe conditions for schlichtness that have stood the test of time and are distinguished by the simplicity of their derivations and the richness of their applications. In the third chapter we give a characterization of the more “fiery” points of the subject: the principles laid down there of constructing sufficient conditions for schlichtness are in our view the most far-reaching. In the fourth chapter we set out in schematic form applications of the results of Chapters I–III, and we mention a few problems. The fifth chapter is concerned with questions that impinge on our theme, but lie outside the main interests of the authors of this article.