Schwarzian derivative, condenser capacities and symmetrization

In the talk we will give a brief survey of results involving the Schwarzian derivative and depending on the geometry of the image of a domain under a holomorphic map. The author's results obtained previously by using the theory of condenser capacity and symmetrization constitute the core of the talk. Inequalities for univalent and multivalent functions are considered both at interior and at boundary points of the domain of definition.
This work was supported by the Russian Science Foundation under grant no. 14-11-00022.