This monograph deals with the application of the method of the extremal metric to the theory of univalent functions. Apart from an introductory chapter in which a brief survey of the development of this theory is given there is therefore no attempt to follow up other methods of treatment. Nevertheless such is the power of the present method that it is possible to include the great majority of known results on univalent functions. It should be mentioned also that the discussion of the method of the extremal metric is directed toward its application to univalent functions, there being no space to present its numerous other applications, particularly to questions of quasiconformal mapping. Also it should be said that there has been no attempt to provide an exhaustive biblio graphy, reference normally being confined to those sources actually quoted in the text. The central theme of our work is the General Coefficient Theorem which contains as special cases a great many of the known results on univalent functions. In a final chapter we give also a number of appli cations of the method of symmetrization. At the time of writing of this monograph the author has been re ceiving support from the National Science Foundation for which he wishes to express his gratitude. His thanks are due also to Sister BARBARA ANN Foos for the use of notes taken at the author's lectures in Geo metric Function Theory at the University of Notre Dame in 1955-1956.

Inhalt
I. Introduction.- Basic definitions. Classical results. Special families. Method of Prawitz. Method of Löwner. Method of the extremal metric. Method of contour integration. Variational method. Multivalent functions. Symmetrization.- II. Modules and Extremal Lengths.- Fundamental definitions. Basic properties of modules. Some special modules. Uniqueness lemmas. Grötzsch's lemmas. Reduced module. Generalizations. An application.- III. Quadratic Differentials.- Definitions. Local structure of the trajectories. Global structure of the trajectories on a finite oriented Riemann surface. The Three Pole Theorem.- IV. The General Coefficient Theorem.- Statement of the General Coefficient Theorem. Differential-Definitions geometric lemmas. Construction of special subsurface. Estimation of the area of its image from above and below. Proof of the fundamental inequality. Discussion of the possibility of equality. Extended Theorem.- V. Canonical Conformal Mappings.- Circular, radial and spiral slit mappings. Parallel slit mappings. Paralic, elliptic and hyperbolic slit mappings. Domains of infinite conectivity.- VI. Applications of the General Coefficient Theorem. Univalent Functions. Proofs of the classical results and extensions. Diameter theorems. Regions of values results for functions in ? (D) and ?, their derivatives and certain coefficients. Regions of values results for functions in 5 and their derivatives. Teichmüller's coefficient results.- VII. Applications of the General Coefficient Theorem. Families of Univalent Functions.- Results on the inner radius for non-overlapping domains. New classes of problems: an example.- VIII. Symmetrization. Multivalent Functions.- Definitions. Geometrical results on symmetrization. Relation to Dirichlet integrals and modules.Uniqueness results for modules. Extension to Riemann domains. Application to multivalent functions.- Authorlndex.