On the ranges of analytic functions

Author:
J. S. Hwang

Journal:
Trans. Amer. Math. Soc. **260** (1980), 623-629

MSC:
Primary 30D40

MathSciNet review:
574804

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Abstract | References | Similar Articles | Additional Information

Abstract: Following Doob, we say that a function analytic in the unit disk *U* has the property if and for some on the unit circle whose measure ,

We recently have solved a problem of Doob by showing that there is an integer such that no function with the property can satisfy

The function

**[1]**F. Bagemihl and W. Seidel,*Koebe arcs and Fatou points of normal functions*, Comment. Math. Helv.**36**(1961), 9–18. MR**0141786****[2]**E. F. Collingwood and A. J. Lohwater,*The theory of cluster sets*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR**0231999****[3]**Joseph L. Doob,*The ranges of analytic functions*, Ann. of Math. (2)**36**(1935), no. 1, 117–126. MR**1503212**, 10.2307/1968668**[4]**G. H. Hardy and J. E. Littlewood,*A further note on Abel's theorem*, Proc. London Math. Soc.**25**(1926), 219-236.**[5]**J. S. Hwang,*The range of a gap series*, Canad. Math. Bull.**18**(1975), no. 5, 753–754. MR**0412396****[6]**-,*On two problems of Doob about the ranges of analytic functions*, Notices Amer. Math. Soc.**25**(1978), A-429.**[7]**J. S. Hwang,*On an extremal property of Doob’s class*, Trans. Amer. Math. Soc.**252**(1979), 393–398. MR**534128**, 10.1090/S0002-9947-1979-0534128-6**[8]**Olli Lehto and K. I. Virtanen,*Boundary behaviour and normal meromorphic functions*, Acta Math.**97**(1957), 47–65. MR**0087746**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0574804-0

Keywords:
The range and analytic function

Article copyright:
© Copyright 1980
American Mathematical Society