Rapidly growing entire functions with three singular values. (English) Zbl 1186.30029

The paper under review deals with the growth behavior of meromorphic functions with finitely many singular values. Let  \(f\)  be a transcendental meromorphic function on \(\mathbb{C}\). A critical point of \(f\) is a point at which the spherical derivative of \(f\) vanishes, and the value of a critical point is called a critical value of \(f\). A point \(a\) in \(\overline{\mathbb{C}}\) is called a singular value of \(f\) if it is either a critical value or an asymptotic value of  \(f\).
The main result in this paper can be stated as follows: Let  \(M(r)\)  be an arbitrary \(\mathbb{R}\)-valued function on \([0,+\infty)\). Then there is an entire function \(f\) with three singular values  \(0,~1\)  and \(\infty\) such that \(T(r,~f)\geq M(r)\)   on \([r_0,+\infty)\) for some \(r_0>0\). The proof of the above theorem is based on a combinatorial construction of a Riemann surface spread over \(\overline{\mathbb{C}}\) that branches over three points.


30D15 Special classes of entire functions of one complex variable and growth estimates
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30F20 Classification theory of Riemann surfaces
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
Full Text: arXiv Euclid


[1] M. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres , Invent. Math. 150 (2002), 127–183. · Zbl 1037.53023
[2] J. W. Cannon, The combinatorial Riemann mapping theorem , Acta Math. 173 (1994), 155–234. · Zbl 0832.30012
[3] P. G. Doyle, Random walk on the Speiser graph of a Riemann surface , Bull. Amer. Math. Soc. (N.S.) 11 (1984), 371–377. · Zbl 0545.60069
[4] P. G. Doyle and J. L. Snell, Random walks and electric networks , Math. Assoc. Amer., Washington, DC, 1984. · Zbl 0583.60065
[5] R. J. Duffin, The extremal length of a network, J. Math. Anal. Appl. 5 (1962), 200–215. · Zbl 0107.43604
[6] A. Eremenko, Transcendental meromorphic functions with three singular values , Illinois J. Math. 48 (2004), 701–709. · Zbl 1070.30010
[7] A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions , Nauka, Moscow, 1970. (In Russian.)
[8] M. Gromov, Hyperbolic manifolds, groups and actions , Ann. Math. Stud., vol. 97, Princeton Univ. Press, 1981, pp. 183–213. · Zbl 0467.53035
[9] W. K. Hayman, Meromorphic functions , Clarendon Press, Oxford, 1964. · Zbl 0115.06203
[10] J. Heinonen, Lectures on analysis on metric spaces , Springer-Verlag, New York, 2001. · Zbl 0985.46008
[11] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry , Acta Math. 181 (1998), 1–61. · Zbl 0915.30018
[12] M. Kanai, Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds , J. Math. Soc. Japan 37 (1985), 391–413. · Zbl 0554.53030
[13] M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds , J. Math. Soc. Japan 38 (1986), 227–238. · Zbl 0577.53031
[14] J. K. Langley, On differential polynomials, fixpoints and critical values of meromorphic functions , Result. Math. 35 (1999), 284–309. · Zbl 0932.30030
[15] J. K. Langley, Critical values of slowly growing meromorphic functions , Comput. Methods Funct. Theory 2 (2002), 537–547. · Zbl 1048.30015
[16] S. Merenkov, Determining biholomorphic type of a manifold using combinatorial and algebraic structures, Ph.D. thesis, 2003.
[17] R. Nevanlinna, Eindeutige analytische Funktionen , Springer-Verlag, Berlin, 1936 (and also 1974). Translated as Analytic functions , Springer-Verlag, 1970.
[18] O. Schramm, Square tilings with prescribed combinatorics , Israel J. Math. 84 (1993), 97–118. · Zbl 0788.05019
[19] A. Speiser, Über Riemannsche Flächen, Comment. Math. Helv. 2 (1930), 254–293. · JFM 56.0987.03
[20] A. Speiser, Probleme aus dem Gebiet der ganzen transzendenten Funktionen , Comment. Math. Helv. 1 (1929), 289–312. · JFM 55.0189.01
[21] S. Stoïlow, Leçons sur les Principes Topologiques de la Théorie des Fonctions Analytiques, Paris, 1956. · Zbl 0072.07604
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