Univalent Functions and Conformal Mapping

I. Introduction.- Basic definitions. Classical results. Special families. Method of Prawitz. Method of Lowner. 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. Grotzsch'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. Teichmuller'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.