Locally univalent functions and the Bloch and Dirichlet norm (Q260296)

The authors obtain estimates for \(|f'(z)|\) and \(|f(z)|\) for functions \(f\) that are analytic in the unit disk \(D\) and satisfy \[ (1-|z|^2)|f''(z)/f'(z)| \leq 1,\quad z \in D, \] (this condition implies the univalence of \(f\)). Moreover they prove that a function \(f\) is univalent in \(D\) under some conditions on the Dirichlet norm of \(\log f'\).