Summary based on review in Mathematical Reviews by H.L. Royden
This book explores the properties of conformal mappings of a plane domain for which the Dirichlet integral of the mappings is finite and the diameter of the image of a set is bounded in terms of the diameter of the image of its boundary. Chapter III studies the Carathéodory theory of boundary correspondence and prime ends. Chapter IV considers sequences of mappings and the Carathéodory theory of domain convergence. Chapter V studies functions on a lattice and harmonic and holomorphic functions. Chapter VI discusses the length-area principle of Grotzsch and some of its applications. Chapter VII deals with the problem of defining a class of functions with a finite Dirichlet integral so that the limits of functions of this class are still in the class.