Abstract
We show that if f and g are transcendental entire functions such that f(g) = g(f), then f satisfies an algebraic differential equation if and only if g does.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. N. Baker, Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1959), 121–163.
D. Bargmann, Iteration holomorpher Funktionen, Dissertation, Christian-Albrechts-Universität zu Kiel, 1996.
W. D. Brownawell, On the factorization of partial differential equations, Canad. J. Math. 39 (1987), 825–834.
J. Clunie, The composition of entire and meromorphic functions, in: H. Shankar (ed.), Mathematical essays dedicated to A. J. Macintyre, Athens, Ohio, Ohio University Press 1970, 75–92.
A. E. Eremenko, Some functional equations connected with the iteration of rational functions (in Russian), Algebra i Analiz 1 (1989), 102–116; translation in: Leningrad Math. J. 1 (1990), 905–919.
P. Fatou, Sur l’itération analytique et les substitutions permutables, J. Math. (9) 2 (1923), 343–384.
F. Gross and C. F. Osgood, A simpler proof of a theorem of Steinmetz, J. Math. Anal. Appl. 143 (1989), 490–496.
F. Gross and C. F. Osgood, An extension of a theorem of Steinmetz, J. Math. Anal. Appl. 156 (1991), 290–294.
F. Gross and C. F. Osgood, Finding all solutions related to Steinmetz’s theorem, J. Math. Anal. Appl. 164 (1992), 417–421.
W. K. Hayman, Meromorphic Functions, Oxford, Clarendon Press, 1964.
T. Kobayashi, Permutability and unique factorizability of certain entire functions, Kodai Math. J. 3 (1980), 8–25.
G. Julia, Mémoire sur la permutabilité des fractions rationnelles, Ann. Sci. Ecole Norm. Sup. (3) 39 (1922), 131–215.
T. W. Ng, Permutable entire functions and their Julia sets, Math. Proc. Cambridge Philos. Soc. 131 (2001), 129–138.
A. Ostrowski, Uber Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), 241–298.
G. Pólya, On an integral function of an integral function, J. London Math. Soc. 1 (1926), 12–15.
J. F. Ritt, Transcendental transcendency of certain functions of Poincaré, Math. Ann. 95 (1925/26), 671–682.
N. Steinmetz, Über faktorisierbare Lösungen gewöhnlicher Differentialgleichungen, Math. Z. 170 (1980), 169–180.
C.-C. Yang and H. Urabe, On permutability of certain entire functions, J. London Math. Soc. (2) 14 (1976), 153–159.
W. Xiong, On the permutability of entire functions, Nanjing Daxue Xuebao Shuxue Bannian Kan 17 (2000), 56–63.
J. H. Zheng, On permutability of periodic entire functions, J. Math. Anal. Appl. 140 (1989), 262–269.
J. H. Zheng and C.-C. Yang, Permutability of entire functions, Kodai Math. J. 15 (1992), 230–235.
J. H. Zheng and C.-C. Yang, On the permutability of entire functions, J. Math. Anal. Appl. 167 (1992), 152–159.
J. H. Zheng and Z. Z. Zhou, Permutability of entire functions satisfying certain differential equations, Tohoku Math. J. (2) 40 (1988), 323–330.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Walter K. Hayman on the occasion of his 80th birthday
Supported by the Alexander von Humboldt Foundation and by the G.I.F., the German-Israeli Foundation for Scientific Research and Development, Grant G-809-234.6/2003.
Rights and permissions
About this article
Cite this article
Bergweiler, W. Permutable Entire Functions Satisfying Algebraic Differential Equations. Comput. Methods Funct. Theory 8, 101–106 (2008). https://doi.org/10.1007/BF03321673
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321673