A variational method for functions of bounded boundary rotation

Abstract

Introduction.In this paper we present a variational method for functions of bounded boundary rotation and solve certain general extremal problems for these functions.The variational method is based upon a general method of G. M. Goluzin [1], and has previously been used by the present author to solve extremal problems for several classes of univalent functions ([7] and [8]).Once the variation formulas are derived, the details of solving the extremal problems are very similar to those in [7] and [8].Theorem 1 here has been obtained independently by Professor J. Pfaltzgraff, who used essentially the same methods.I wish to thank Professor Pfaltzgraff for providing me with his unpublished work.Those functions of bounded boundary rotation which are also univalent form a subclass which has recently been studied by M. M. Schiffer and O. Tammi [9].They use a different (though closely related) variational method.We shall compare their results with ours.