4 - Symmetrization

Summary

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 D_f 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 D_f 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].