On the univalency of certain classes of analytic functions

In Chapter I we begin by considering a theorem of J. Dieudonne on the minimum radius of starlikeness of a class of analytic functions. We give a simple new proof of this theorem. By this new proof also we find the minimum radius of univalence of this class and we determine all the cases which give the minimum radius of univalence and the minimum radius of starlikeness. We then use a method similar to that in this new proof to obtain the minimum radius of univalence and the minimum radius of starlikeness of some other classes of analytic functions. For each class we determine all the cases giving the minimum radius of univalence and the minimum radius of starlikeness. Then we give some similar results for the minimum radius of convexity. In Chapter II first we deal with Heawood's Lemma which was established and used by P.J. Heawood to prove the theorem known as the Grace-Heawood Theorem. The same lemma was used by S. Kakeya in the proof of another theorem. We show that Heawood's Lemma is false and we give new proofs of these results. Then for some special cases we improve the value of the radius of univalencegiven by Kakeya's Theorem. In this connection we firstgive L.N. Cakalov's result and then we obtain some improvements of his result. In Appendix I we give some examples related with Chapter I and Chapter II. In Appendix II we give an example which shows that there is an error in a paper by M. Robertson.