Abstract
On a Stein manifold, we obtain generic versions of two covering theorems of Valiron and Birkhoff respectively. Namely, we show that most equidimensional holomorphic mappings into complex Euclidean space contain arbitrarily large ‘schlicht’ balls in their images. Moreover, most global holomorphic functions have a universality property which consists in approximately reproducing all local holomorphic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. A. Aizenberg and A. P. Yuzhakov, Integral Representation and Residues in Multidimensional Complex Analysis, American Mathematical Society, Providence, RI, 1983.
G. D. Birkhoff, Démonstration d’un théorème élémentaire sur les fonctions entières, C.R. Acad. Sci. Paris 189 (1929), 473–475.
M. Bonk, On Bloch’s constant, Proc. Am. Math. Soc. 110 no.4 (1990), 889–894.
H. Chen and P. M. Gauthier, On Bloch’s constant, J. Anal. Math. 69 (1996), 275–291.
F. Deutsch, Simultaneous interpolation and approximation in topological linear spaces, SIAM J. Appl. Math. 14 (1966), 1180–1190.
A. Eremenko and M. Ju. Ljubich, Examples of entire functions with pathological dynamics, J. London Math. Soc.(2) 36 no.3 (1987), 458–468.
P. M. Gauthier, Tangential approximation and universality. Izv. Akad. Nauk Arm. SSR, Mat. 36 (2001), 38–48.
E. Kallin, Polynomial convexity: The three spheres problem, in: Proc. Conf Complex Analysis (Minneapolis, 1964) pp. 301–304. Springer, Berlin.
L. Kaup and B. Kaup, Holomorphic Functions of Several Variables, Walter de Gruyter & Co., Berlin, 1983.
A. A. Nersesjan, A uniform approximation with simultaneous interpolation by analytic functions, (Russian) Izv. Akad. Nauk Arm. SSR, Mat. 15 no.4 (1980), 249–257.
Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, American Mathematical Society, Providence, RI, 1989.
J.-P. Rosay and W. Rudin, Holomorphic maps from Cn to Cn, Trans. Am. Math. Soc. 310 no.1 (1988), 47–86
G. Valiron, Sur les théorèmes de MM. Bloch, Landau, Montel et Schottky, C. R. 183 (1926), 728–730.
C. Wagschal, Fonctions holomorphes équations différentielles, Hermann, Paris, 2003.
Ch. Xiong, Lower bound of Bloch’s constant. J. Nanjing Univ., Math. Biq.15 no.2 (1998), 174–179
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by NSERC (Canada) and project no. 800415 University of Isfahan (Iran).
Rights and permissions
About this article
Cite this article
Gauthier, P.M., Pouryayevali, M.R. Covering Properties of Most Entire Functions on Stein Manifolds. Comput. Methods Funct. Theory 5, 223–235 (2005). https://doi.org/10.1007/BF03321095
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03321095