On the Schwarz reflection principle

Author:
J. S. Hwang

Journal:
Trans. Amer. Math. Soc. **272** (1982), 711-719

MSC:
Primary 30D40

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662062-X

MathSciNet review:
662062

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently, we have solved a long outstanding problem of A. J. Lohwater (1953) by showing that if is meromorphic in whose radial limits have modulus 1 for almost all points on an arc of , and if is a singular point of on , then every value of modulus 1 which is not in the range of at is an asymptotic value of at some point of each subarc of containing the point .

Lohwater proved this theorem for functions of bounded characteristic and he made a comment that his method is not, in general, applicable to functions of unbounded characteristic. In this paper, we shall present an alternative proof of the above theorem based on the very method of Lohwater.

**[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 Univ. Press, London, 1966. MR**0231999 (38:325)****[3]**J. S. Hwang,*On an extremal property of Doob's class*, Trans. Amer. Math. Soc.**252**(1979), 393-398. MR**534128 (80i:30057)****[4]**-,*On a problem of Lohwater about the asymptotic behaviour of Nevanlinna's class*, Proc. Amer. Math. Soc.**81**(1981), 538-540. MR**601724 (82h:30031)****[5]**O. Lehto and K. I. Virtanen,*Boundary behaviour and normal meromorphic functions*, Acta Math.**97**(1957), 47-65. MR**0087746 (19:403f)****[6]**A. J. Lohwater,*On the Schwartz reflection principle*, Michigan Math. J.**2**(1953-1954), 151-156. MR**0068624 (16:914c)****[7]**W. Seidel,*On the cluster values of analytic functions*, Trans. Amer. Math. Soc.**34**(1932), 1-21. MR**1501628****[8]**-,*On the distribution of values of bounded analytic functions*, Trans. Amer. Math. Soc.**36**(1934), 201-226. MR**1501738****[9]**M. Tsuji,*Potential theory in modern function theory*, Maruzen, Tokyo, 1959. MR**0114894 (22:5712)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30D40

Retrieve articles in all journals with MSC: 30D40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0662062-X

Keywords:
Asymptotic behaviour,
bounded characteristic,
reflection principle,
Seidel's class

Article copyright:
© Copyright 1982
American Mathematical Society