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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.

MSC:

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)
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