Nonnormal Dirichlet Quotients and Nonnormal Blaschke Quotients

Abstract

There exists a nonnormal meromorphic function ${f_1}/{f_2}$ in $U = \{ |z| < 1\}$, where ${f_1}$ and ${f_2}$ both are holomorphic functions with finite Dirichlet integrals in U. For each $0 < \alpha < 1$, there exists a nonnormal meromorphic function ${B_1}/{B_2}$ in U, where ${B_1}$ and ${B_2}$ both are Blaschke products with finite $\alpha$-weighted Dirichlet integrals in U.