A problem on the Bloch norm of functions in Doob's class

Author:
J. S. Hwang

Journal:
Proc. Amer. Math. Soc. **95** (1985), 554-556

MSC:
Primary 30C45; Secondary 30C80

DOI:
https://doi.org/10.1090/S0002-9939-1985-0810162-1

MathSciNet review:
810162

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Abstract | References | Similar Articles | Additional Information

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

**[1]**E. F. Collingwood and A. J. Lohwater,*The theory of cluster sets*, Cambridge Univ. Press, London and New York, 1966. MR**0231999 (38:325)****[2]**J. L. Doob,*The ranges of analytic functions*, Ann. of Math. (2)**36**(1935), 117-126. MR**1503212****[3]**S. Dragosh and D. C. Rung,*Normal functions bounded on arcs and a proof of the Gross cluster-value theorem*, Hiroshima Math. J.**9**(1979), 303-312. MR**535513 (80j:30043)****[4]**E. Hille,*Analytic function theory*, vol. II, Ginn, Boston, Mass., 1962. MR**0201608 (34:1490)****[5]**J. S. Hwang and D. C. Rung,*Proof of a conjecture of Doob*, Proc. Amer. Math. Soc.**75**(1979), 231-234. MR**532142 (81i:30061)****[6]**-,*An improved estimate for the Bloch norm of functions in Doob's class*, Proc. Amer. Math. Soc.**80**(1980), 406-410. MR**580994 (81i:30058)****[7]**J. S. Hwang,*On an extremal property of Doob's class*, Trans. Amer. Math. Soc.**252**(1979), 393-398. MR**534128 (80i:30057)****[8]**W. Seidel,*On the distribution of values of bounded analytic functions*, Trans. Amer. Math. Soc.**36**(1934), 201-226. MR**1501738**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1985-0810162-1

Keywords:
Bloch norm,
Doob's class,
maximum principle

Article copyright:
© Copyright 1985
American Mathematical Society