Univalent functions in Hardy spaces in terms of the growth of arc-length
Pommerenke proved in 1962 that for \(f\) univalent in the unit disk and \(0<p<2\), \(f\in H^p\) if and only if \[ \int_0^1M_1^p(r,f')\,dr < \infty. \] In this paper, the authors prove that this result continues to be true for \(p\) strictly larger than \(2\), but is false for large \(p\). In addition, it turns out that the result is true for all \(p\) when \(f\) is restricted to the class of close-to-convex functions. Finally, the authors discuss the membership of univalent functions in some related spaces of Dirichlet type.
scientific article; zbMATH DE number 5624077
scientific article