2016/08/02 by Schippers, Eric
#30C50 #30C55 #30C75 #30C85 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1608.00790
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 ⊆ 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 (∂ 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.