Transcendental meromorphic functions with three singular values

Abstract

Every transcendental meromorphic function f in the plane, which has only three critical values, satisfies, and this estimate is best possible.

The Hausdorff dimension of Julia sets of meromorphic functions in the Speiser class

“…defines a meromorphic function h m ∈ S 3 . Similar examples were already considered by Teichmüller [46, p. 734], and later by Bank and Kaufman [7, Section 5], Langley [31, Section 2] and Eremenko [23].…”

“…and note that this defines a meromorphic function F ∈ S 3 . Examples of this type were previously considered in [41, p. 734], [5, Section 5], [30, Section 2], [23] and [10]. We note that the points z k := π/2 + ikπ/2 are critical points of H. It follows that the points…”

A survey of some results after 1970

“…This constant c(f ) can be arbitrarily small, but in the case that f has only three critical and asymptotic values, we have [49] c(f ) ≥ √ 3/(2π) and this is best possible. On the other hand, there are no restrictions from above on the growth of functions of class S [113].…”

Rapidly growing entire functions with three singular values

“…The question of slowest possible growth of the Nevanlinna characteristic for meromorphic functions with finitely many singular values has been studied in recent years, notably by A. Eremenko [6], and J. Langley [14], [15]. In particular it was proved that if f is a transcendental meromorphic function with three singular values, then…”