Abstract
Let\(S_ \propto ( \propto \geqq 0)\) be the set of normalized (see (1.2)) functions f holomorphic in D:|z|<1 with\(f''(z)/f'(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\), and let
be the set of normalized (see (1.6)) functions f meromorphic in D with the Schwarzian derivative\(\left\{ {f,z} \right\} = 0((1 - \left| z \right|^2 )^{ - \propto } )\). We shall show that some topological properties of\(S_ \propto\) and
, and of subsets of them, follow from those of the weighted H∞ space\(H_ \propto ^\infty\), consisting of functions f holomorphic in D with\(f(z) = 0((1 - \left| z \right|^2 )^{ - \propto } )\), and those of subsets of\(H_ \propto ^\infty\). The set S1 is denoted by X in [3] and [4].
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References
AHLFORS, L. V.: Quasiconformal reflections. Acta Math.109, 291–301 (1963)
ANDERSON, J. M., CLUNIE, J. and POMMERENKE CH.: On Bloch functions and normal functions. J. reine angew. Math.270, 12–37 (1974)
CAMPBELL, D. M.: Uniform convergence on compacta and locally univalent analytic functions of finite order. Monatsh. Math.77, 21–23 (1973)
CAMPBELL, D. M., CIMA, J. A. and PFALTZGRAFF, J. A.: Linear spaces and linear-invariant families of locally univalent analytic functions. Manuscripta Math.4, 1–30 (1971)
DUREN, P. L., SHAPIRO, H. S. and SHIELDS, A. L.: Singular measures and domains not of Smirnov type. Duke Math. J.33, 247–254 (1966)
DUREN, P. L. and LEHTO, O.: Schwarzian derivatives and homeomorphic extensions. Ann. Acad. Sci. Fenn. Ser. A I. Math.477 1–11 (1970)
HORNICH, H.: Ein Banachraum analytischer Funktionen in Zusammenhang mit den schlichten Funktionen. Monatsh. Math.73, 36–45 (1969)
HORNICH, H.: Über einen Banachraum analytischer Funktionen. Manuscripta Math.1, 79–86 (1969)
KIM, Y. J. and MERKES, E. P.: On certain convex sets in the space of locally schlicht functions. Trans. Amer. Math. Soc.196, 217–224 (1974)
LAPPAN, P.: A non-normal locally uniformly univalent function. Bull. London Math. Soc.5, 291–294 (1973)
NEHARI, Z.: The Schwarzian derivative and schlicht functions. Bull. Amer. Math. Soc.55, 545–551 (1949)
POMMERENKE, CH.: On Bloch functions. J. London Math. Soc. (2)2, 689–695 (1970)
POMMERENKE, CH.: Linear-invariante Familien analytischer Funktionen I. Math. Ann.155, 108–154 (1964)
YAMASHITA, S.: The derivative of a holomorphic function in the disk. Michigan Math. J.21, 129–132 (1974)
YAMASHITA, S.: On normal meromorphic functions. Math. Zeits.,141, 139–145 (1975)
YAMASHITA, S.: Local schlichtness of a function meromorphic in the disk. Math. Nachr., to be published
YAMASHITA, S.: Almost locally univalent functions. Monatsh. Math., to be published
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Yamashita, S. Banach spaces of locally Schlicht functions with the Hornich operations. Manuscripta Math 16, 261–275 (1975). https://doi.org/10.1007/BF01164428
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DOI: https://doi.org/10.1007/BF01164428