On the ranges of analytic functions
Author:
J. S. Hwang
Journal:
Trans. Amer. Math. Soc. 260 (1980), 623629
MSC:
Primary 30D40
MathSciNet review:
574804
Fulltext PDF Free Access
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 shows that the condition is necessary and cannot be replaced by , for . Naturally, we may ask whether this can be replaced by , for ? The answer turns out to be yes, when , where .
 [1]
F.
Bagemihl and W.
Seidel, Koebe arcs and Fatou points of normal functions,
Comment. Math. Helv. 36 (1961), 9–18. MR 0141786
(25 #5183)
 [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
(38 #325)
 [3]
Joseph
L. Doob, The ranges of analytic functions, Ann. of Math. (2)
36 (1935), no. 1, 117–126. MR
1503212, http://dx.doi.org/10.2307/1968668
 [4]
G. H. Hardy and J. E. Littlewood, A further note on Abel's theorem, Proc. London Math. Soc. 25 (1926), 219236.
 [5]
J.
S. Hwang, The range of a gap series, Canad. Math. Bull.
18 (1975), no. 5, 753–754. MR 0412396
(54 #522)
 [6]
, On two problems of Doob about the ranges of analytic functions, Notices Amer. Math. Soc. 25 (1978), A429.
 [7]
J.
S. Hwang, On an extremal property of
Doob’s class, Trans. Amer. Math. Soc.
252 (1979),
393–398. MR
534128 (80i:30057), http://dx.doi.org/10.1090/S00029947197905341286
 [8]
Olli
Lehto and K.
I. Virtanen, Boundary behaviour and normal meromorphic
functions, Acta Math. 97 (1957), 47–65. MR 0087746
(19,403f)
 [1]
 F. Bagemihl and W. Seidel, Koebe arcs and Fatou points of normal functions, Comment. Math. Helv. 36 (1961), 918. MR 0141786 (25:5183)
 [2]
 E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Univ. Press, London and New York, 1966. MR 0231999 (38:325)
 [3]
 J. L. Doob, The ranges of analytic functions, Ann. of Math. 36 (1935), 117126. MR 1503212
 [4]
 G. H. Hardy and J. E. Littlewood, A further note on Abel's theorem, Proc. London Math. Soc. 25 (1926), 219236.
 [5]
 J. S. Hwang, The range of a gap series, Canad. Math. Bull. 18 (1975), 753754. MR 0412396 (54:522)
 [6]
 , On two problems of Doob about the ranges of analytic functions, Notices Amer. Math. Soc. 25 (1978), A429.
 [7]
 , On an extremal property of Doob's class, Trans. Amer. Math. Soc. 252 (1979), 393398. MR 534128 (80i:30057)
 [8]
 O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 4765. MR 0087746 (19:403f)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
30D40
Retrieve articles in all journals
with MSC:
30D40
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198005748040
PII:
S 00029947(1980)05748040
Keywords:
The range and analytic function
Article copyright:
© Copyright 1980
American Mathematical Society
