This is the first systematic presentation of the capacitory approach and symmetrization in the context of complex analysis. The content of the book is original the main part has not been covered by existing textbooks and monographs. After an introduction to the theory of condenser capacities in the plane, the monotonicity of the capacity under various special transformations (polarization, Gonchar transformation, averaging transformations and others) is established, followed by various types of symmetrization which are one of the main objects of the book. By using symmetrization principles, some metric properties of compact sets are obtained and some extremal decomposition problems are solved. Moreover, the classical and present facts for univalent and multivalent meromorphic functions are proven.

This book will be a valuable source for current and future researchers in various branches of complex analysis and potential theory.

Preface.- 1.Conformal capacity.- 2. Asymptotics of the condenser capacity when one of the plate degenerates.- Special transformations.- 4. Symmetrization.- 5. Metric properties of sets and condensers.- 6. Problems of extremal decomposition.- 7. Univalent functions.- 8. Multivalent functions.- Appendices: A1. Dirichlet principle.- A2. Uniqueness theorem for contracting mapping.- A3. On separating transformation of sets and condensers.- A4. On conservation of reduced moduli under geometric trans-formation of domains.- A5. Quadratic differentials.- A6. Unsolved problems.- References.- Basic notations.- Subject index.