Hostname: page-component-68945f75b7-s5tfc Total loading time: 0 Render date: 2024-09-02T16:16:30.165Z Has data issue: false hasContentIssue false
  • English
  • Français

On the fixpoints of composite entire functions of finite order

Published online by Cambridge University Press:  14 November 2011

J. K. Langley
J. K. Langley
Affiliation:
Department of Mathematics, University of Nottingham, NG7 2RD, U.K.
Get access Rights & Permissions [Opens in a new window]

Extract

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 5 , 1994 , pp. 995 - 1001
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Anderson, J. M., Baker, I. N. and Clunie, J.. The distribution of values of certain entire and meromorphic functions. Math. Z. 178 (1981), 509–525.CrossRefGoogle Scholar
2Barry, P. D.. On a theorem of Besicovitch. Quart. J. Math. Oxford Ser. (2) 14 (1963), 293–302.CrossRefGoogle Scholar
3Bergweiler, W.. Proof of a conjecture of Gross concerning fixpoints. Math. Z. 204 (1990), 381–390.CrossRefGoogle Scholar
4Bergweiler, W.. Periodische Punkte bei der Iteration ganzer Funktionen (Habilitationsschrift, Aachen, 1991).Google Scholar
5Bergweiler, W.. Periodic points of entire functions, proof of a conjecture of Baker. Complex Variables Theory Appl. 17 (1991), 57–72.Google Scholar
6Bergweiler, W. and Yang, C. C.. On the value distribution of composite meromorphic functions. Bull. London Math. Soc. 25 (1993), 357–361.CrossRefGoogle Scholar
7Clunie, J. G.. The composition of entire and meromorphic functions. In Mathematical Essays Dedicated to A. J. Macintyre, 75–92 (Athens, Ohio: Ohio University Press, 1970).Google Scholar
8Drasin, D.. Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two. Acta Math. 158 (1987), 1–94.CrossRefGoogle Scholar
9Edrei, A. and Fuchs, W. H.J.. Valeurs déficientes et valeurs asymptotiques des fonctions méromorphes. Comment. Math. Helv. 33 (1959), 258–295.CrossRefGoogle Scholar
10Eremenko, A.. A new proof of Drasin's Theorem on meromorphic functions of finite order with maximal deficiency sum, I and II. Teor. Funktsĭil, Funktsional. Anal, i Prilozhen. 51 (1989), 107–116; 52 (1989), 69–77 (in Russian). English translation: J. Soviet Math. 52 (1990), 3522–3529; 52 (1990), 3397–3403.Google Scholar
11Eremenko, A.. Meromorphic functions with small ramification. Indiana Univ. Math. J. (to appear).Google Scholar
12Goldstein, R.. On factorization of certain entire functions. J. London Math. Soc. (2) 2 (1970), 221–224.CrossRefGoogle Scholar
13Goldstein, R.. On factorization of certain entire functions. II. Proc. London Math. Soc. (3) 22 (1971), 483–506.CrossRefGoogle Scholar
14Gross, F.. Factorization of Meromorphic Functions (Washington, D.C.: Mathematics Research Center, Naval Research Lab., 1972).Google Scholar
15Hayman, W. K.. Meromorphic Functions (Oxford: Clarendon Press, 1964).Google Scholar
16Katajamäki, K., Kinnunen, L. and Laine, I.. On the value distribution of composite entire functions. Complex Variables Theory Appl. (to appear).Google Scholar
17Katajamäki, K., Kinnunen, L. and Laine, I.. On the value distribution of some composite meromorphic functions. Bull. London Math. Soc. 25 (1993), 445–452.CrossRefGoogle Scholar
18Langley, J. K.. On the deficiencies of composite entire functions. Proc. Edinburgh Math. Soc. 36 (1992), 151–164.CrossRefGoogle Scholar
19Tsuji, M.. Potential Theory in Modern Function Theory (Tokyo: Maruzen, 1959).Google Scholar
20Yang, C. C. and Zheng, J. H.. Further results on fixpoints and zeros of entire functions (preprint).Google Scholar