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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Higher dimensional generalizations of the Bloch constant and their lower bounds

Author: Kyong T. Hahn
Journal: Trans. Amer. Math. Soc. 179 (1973), 263-274
MSC: Primary 32A30; Secondary 32H99
MathSciNet review: 0325994
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Abstract: A higher dimensional generalization of the classical Bloch theorem depends in an essential way on the ``boundedness'' of the family of holomorphic mappings considered. In this paper the author considers two types of such ``bounded'' families and obtains explicit lower bounds of the generalized Bloch constants of these families on the hyperball in the space $ {{\mathbf{C}}^n}$ in terms of universal constants which characterize the families.

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Keywords: Classical Bloch theorem, generalized Bloch constant, generalized Koebe constant, holomorphic mapping, the smallest and the largest characteristic values, univalent mapping, quasiconformal holomorphic mapping of order K, hyperball, lower bounds of the Bloch constants, Schwarz lemma, Koebe function
Article copyright: © Copyright 1973 American Mathematical Society