Introduction In this chapter we develop the theory of symmetrization in the form due to Pólya and Szegö [1951] as far as it is necessary for our function-theoretic applications.

Given a domain D, we can, by certain types of lateral displacement called symmetrization, transform D into a new domain D* having some aspects of symmetry. The precise definition will be given in § 4.5. Pólya and Szegö showed that while area for instance remains invariant under symmetrization, various domain constants such as capacity, inner radius, principal frequency, torsional rigidity, etc., behave in a monotonic manner.

We shall here prove this result for the first two of these concepts in order to deduce Theorem 4.9, the principle of symmetrization. If f(z) = a0 + a1z + … is regular in |z| < 1, and something is known about the domain Df of values assumed by f(z), this principle allows us to assert that in certain circumstances |a1| will be maximal when f(z) is univalent and Df symmetrical. Applications of this result will be given in Sections 4.10–4.12. Some of these will in turn form the basis of further studies of p-valent functions in Chapter 5. Some of these results can also be proved in another manner by a consideration of the transfinite diameter (Hayman [1951]). The chapter ends with a recent proof of Bloch's Theorem by Bonk [1990].