A conjecture of L. Carleson and applications (Q1803924)

For a meromorphic function \(f\) in \(\{| z|<1\}\) let \(A(r,f)\) be the spherical area of the image of \(\{| z|<r\}\) by \(f\). Let \(T_ \alpha\) be the class of functions for which \[ \int^ 1_ 0 A(r,f)(1- r)^{-\alpha} dr< \infty,\qquad 0\leq \alpha<1. \] \(T_ 1\) is the class of functions \(f\) with the property, that \(A(r,f)\) is bounded. Thus, \(T_ 0\) is the usual Nevanlinna class but for \(\alpha>0\), \(T_ \alpha\) does not contain all bounded analytic functions. Carleson asked whether each function in \(T_ \alpha\) is the quotient of two bounded functions in the same class and this question is answered affirmatively in the paper for more general classes of meromorphic functions in the disc. The technique used for the proof is then applied to study the inner-outer factorization in these classes and to prove that invertible elements in weighted Dirichlet spaces are cyclic for multiplication by \(z\), i.e. their polynomial multiples are dense in the space.