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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0529210-9

MathSciNet review:
529210

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Abstract: The Bloch space is the space of functions *f* analytic in the unit disc *D* such that is bounded. It is shown that 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.

**[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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DOI:
https://doi.org/10.1090/S0002-9939-1979-0529210-9

Article copyright:
© Copyright 1979
American Mathematical Society