On the Schwarz derivative, the Bloch space and the Dirichlet space

It is well known the connection between the growth of the Schwarzian with both the univalence [see Beardon and Gehring (Comment Math Helv 55: 50–64, 1980), Nehari (Bull Am Math Soc 55:545–551, 1949), Ovesea (Novi Sad J Math 26(1):69–76, 1996)] and the quasiconformal extension of the function [see Ahlfors and Weill (Proc Am Math Soc 13:975–978, 1962), Osgood (Old and new on the Schwarzian derivative, Quasiconformal mappings and analysis. Springer, New York, 1998)]. This work shows that previous relationships have geometrical interpretations when the Schwarzian is applied on the Bloch space and on the Dirichlet space. These interpretations are given in terms of a family of three-dimensional cones. Even more, these function spaces allow us to obtain Möbius invariant properties related to the norm induced by the Schwarzian among other consequences.