A problem on the Bloch norm of functions in Doob's class
Author:
J. S. Hwang
Journal:
Proc. Amer. Math. Soc. 95 (1985), 554556
MSC:
Primary 30C45; Secondary 30C80
MathSciNet review:
810162
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Abstract: Let denote the unit disc and denote the unit circle both in the complex plane. Define the Doob's class , , as all holomorphic functions on satisfying (1) , and (2) for some arc with arclength , for all , . Recently the author and Rung [6] proved a conjecture of Doob made in 1935 by showing that the norm We then conjecture that the result should be true if the arc is replaced by a finite union of arcs whose total length is at least . In this paper, we answer this problem. It turns out to be surprising that the answer depends on the connectivity of the union, namely, the answer is no for the disconnected case, but yes for the connected one.
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 J. L. Doob, The ranges of analytic functions, Ann. of Math. (2) 36 (1935), 117126. MR 1503212
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 J. S. Hwang and D. C. Rung, Proof of a conjecture of Doob, Proc. Amer. Math. Soc. 75 (1979), 231234. MR 532142 (81i:30061)
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 , An improved estimate for the Bloch norm of functions in Doob's class, Proc. Amer. Math. Soc. 80 (1980), 406410. MR 580994 (81i:30058)
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 W. Seidel, On the distribution of values of bounded analytic functions, Trans. Amer. Math. Soc. 36 (1934), 201226. MR 1501738
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198508101621
PII:
S 00029939(1985)08101621
Keywords:
Bloch norm,
Doob's class,
maximum principle
Article copyright:
© Copyright 1985 American Mathematical Society
