Symmetrization. Multivalent Functions

Abstract

The General Coefficient Theorem is somewhat restricted in its applications by the requirement it imposes on the functions of the admissible family {f} of carrying poles of the quadratic differential interior to the domains of the admissible family {⊿} into themselves. As we have seen, when ℜ is the sphere in some cases this can be arranged by auxiliary conformai transformations of the whole sphere. In other cases where this is not possible the same effect can be obtained by the method of symmetrization. This method also permits the extension of many results for univalent functions to the case fo multivalent functions. Of course one cannot use the General Coefficient Theorem directly in these situations but Teichmüller’s principle again provides an associated quadratic differential.