Conformal invariants associated with quadratic differentials

Abstract

Z. Nehari developed a general technique for obtaining inequalities for conformal maps and domain functions from contour integrals and the Dirichlet principle. Given a harmonic function with singularity on a domain $R$, it associates a monotonic functional of subdomains $D \subseteq R$. In the case that $R$ is conformally equivalent to a disk, we extend Nehari's method by associating a functional to any quadratic differential on $R$ with specified singularities. Nehari's method corresponds to the special case that the quadratic differential is of the form $(\partial q)^2$ for a singular harmonic function $q$ on $R$. Besides being more general, our formulation is conformally invariant, and has a particularly elegant equality statement. As an application we give a one-parameter family of monotonic, conformally invariant functionals which correspond to growth theorems for bounded univalent functions. These generalize and interpolate the Pick growth theorems, which appear in a conformally invariant form equivalent to a two-point distortion theorem of W. Ma and D. Minda.