A continuation of mathematical research aimed at the development of a comprehensive and unified theory of symmetrization will be carried out in this project. The application of symmetry to problems in mathematical analysis represents one of the most powerful paradigms for the resolution of extremal problems in very general contexts. It is often the case that fundamental inequalities reach their extreme values under conditions of symmetry and great regularity. The isoperimetric inequality is perhaps the best example of such an event. This work seeks to exploit a recently discovered master inequality; an inequality containing some of the most basic relationships used in analysis. These are inequalities relating integrals of gradients, triple convolutions and the Laplace operator, three integral inequalities which play essential roles in the theory of differential equations, potential theory and the theory of analytic functions - to name a few. Specific applications will touch on the problem of the Bloch and Landau constants from function theory. These are measures of the largest disc in the range of (normalized) analytic functions. Another question which may yet be resolved by symmetrization is that of the lowest eigenvalue of a clamped plate. The analogous question for the vibrating membrane has been known for years. In the former case (the biharmonic equation governs), there is a delicate matter involving the change of sign in the lowest eigenvalue which must first be resolved. Extremal problems for complex polynomials, integrability of powers of the derivatives of confomal maps and area distortion of quasiconformal maps will also be treated in this project.
