10.1007/s40096-020-00334-9

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

  • J. Oscar González Cervantes*1,
  1. Instituto Politécnico Nacional, E.S.F.M - I.P.N., Mexico City, 07338, MX

Published in Issue 2020-05-29

How to Cite

González Cervantes, J. O. (2020). On the Schwarz derivative, the Bloch space and the Dirichlet space. Mathematical Sciences, 14(3 (September 2020). https://doi.org/10.1007/s40096-020-00334-9
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Abstract

Abstract 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.

Keywords

  • Schwarz derivative,
  • Bloch space,
  • Dirichlet space

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