Condenser capacities and majorization principles in the geometric function theory of a complex variable

This survey paper is devoted to applications of potential theory to some extremal problems of the geometric function theory of a complex variable. In particular, we present variational principles of conformal mappings that are derived from the properties of generalized condensers and symmetrization in a unified way. The variations of the Robin functions under deformation of a domain or a portion of its boundary are considered. Applications of condensers and majorization principles include distortion theorems for holomorphic functions, covering theorem for $p$-valent functions in a circular annulus, Bernstein-type inequalities for rational functions with prescribed poles, polynomial inequalities and more.