TY - JOUR
T1 - On the Schwarz derivative, the Bloch space and the Dirichlet space
AU - González Cervantes, J. Oscar
N1 - Publisher Copyright:
© 2020, Islamic Azad University.
PY - 2020/9
Y1 - 2020/9
N2 - 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.
AB - 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.
KW - Bloch space
KW - Dirichlet space
KW - Schwarz derivative
UR - http://www.scopus.com/inward/record.url?scp=85119500720&partnerID=8YFLogxK
U2 - 10.1007/s40096-020-00334-9
DO - 10.1007/s40096-020-00334-9
M3 - Artículo
AN - SCOPUS:85119500720
SN - 2008-1359
VL - 14
SP - 235
EP - 240
JO - Mathematical Sciences
JF - Mathematical Sciences
IS - 3
ER -