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An extremal property of the Bloch space

Authors: Lee A. Rubel and Richard M. Timoney
Journal: Proc. Amer. Math. Soc. 75 (1979), 45-49
MSC: Primary 30D99; Secondary 32A10, 46E15
MathSciNet review: 529210
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Abstract: The Bloch space $ \mathcal{B}$ is the space of functions f analytic in the unit disc D such that $ \vert f'(z)\vert(1 - \vert z{\vert^2})$ is bounded. It is shown that $ \mathcal{B}$ is the largest Möbius-invariant linear space of analytic functions that can be equipped with a Möbius-invariant seminorm in such a way that there is at least one ``decent'' continuous linear functional on the space. The term ``decent'' has a simple and precise definition.

References [Enhancements On Off] (What's this?)

  • [1] J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 0361090
  • [2] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • [3] L. A. Rubel and B. A. Taylor, Functional analysis proofs of some theorems in function theory, Amer. Math. Monthly 76 (1969), 483–489; correction, ibid. 77 (1969), 58. MR 0247115
  • [4] Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
  • [5] R. M. Timoney, Bloch functions in several complex variables, Thesis, Univ. of Illinois at Urbana-Champaign, 1978.

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Article copyright: © Copyright 1979 American Mathematical Society