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.

中文翻译：

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关于Schwarz导数，Bloch空间和Dirichlet空间

众所周知，Schwarzian的成长与单性之间的联系[参见Beardon和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）]和该函数的拟保形扩展[请参见Ahlfors和Weill（Proc Am Math Soc 13：975-978，1962），Osgood（旧的和新的Schwarzian导数，准同形映射和分析，Springer，纽约，1998年]。这项工作表明，将Schwarzian应用到Bloch空间和Dirichlet空间时，先前的关系具有几何解释。这些解释是根据三维圆锥体系列给出的。更重要的是，这些函数空间使我们可以获得与Schwarzian归纳的范数有关的Möbius不变性质以及其他后果。