Rapidly growing entire functions with three singular values

Merenkov, Sergei

doi:10.1215/ijm/1248355345

Cited by 4 publications

(3 citation statements)

References 17 publications

(22 reference statements)

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“…Drasin [Dra07] and Merenkov [Mer08] have constructed maps of this class that have irregular and arbitrarily fast growth, respectively. More recently, Bishop (1) every connected component U of F (f ) is unbounded, and ∂U is not locally connected at any finite point, or (2) every connected component of F (f ) is a bounded quasidisc.…”

Section: Theorem (Bounded Fatou Components)mentioning

confidence: 99%

“…Drasin [Dra07] and Merenkov [Mer08] have constructed maps of this class that have irregular and arbitrarily fast growth, respectively. More recently, Bishop [Bis15] has described a method for constructing functions with no asymptotic values, two critical values and only simple critical points, having essentially arbitrary prescribed behaviour near infinity.…”

Section: Maps With Two Critical Valuesmentioning

confidence: 99%

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Hyperbolic entire functions with bounded Fatou components

Bergweiler¹,

Fagella²,

Rempe³

2015

Comment. Math. Helv.

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We show that an invariant Fatou component of a hyperbolic transcendental entire function is a Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values.

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Section: Theorem (Bounded Fatou Components)mentioning

confidence: 99%

Section: Maps With Two Critical Valuesmentioning

confidence: 99%

Hyperbolic entire functions with bounded Fatou components

Bergweiler¹,

Fagella²,

Rempe³

2015

Comment. Math. Helv.

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“…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].…”

Section: Chapter VIImentioning

confidence: 99%

A survey of some results after 1970

2008

Translations of Mathematical Monographs

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The literature on meromorphic functions 1 is very large. There is a comprehensive survey [62] that contains everything that was reviewed on the topic in the Soviet "Referativnyi Zhurnal" in 1953-1970, and a later large survey [67]. More recent surveys [30], [80] and [48] are shorter and have narrower scope. Some books on specific topics in the theory of meromorphic functions published after 1970 are [18], [20], [79], [100], [137], [134], [138], [162]. A survey of the fast developing subject of iteration of meromorphic functions is [7]. Here we give a short survey of some results which are closely related to the problems considered in this book. Thus we do not include many important topics, like geometric theory of meromorphic functions, iteration, composition, differential and functional equations, normal families, Borel and Julia directions, uniqueness theorems, and most regrettably, holomorphic curves and quasiregular mappings.

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Entire functions arising from trees

Cui

2021

Sci. China Math.

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Rapidly growing entire functions with three singular values

Abstract: We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic dominates an arbitrarily prescribed function.

Cited by 4 publications

References 17 publications

Hyperbolic entire functions with bounded Fatou components

Hyperbolic entire functions with bounded Fatou components

A survey of some results after 1970

Entire functions arising from trees

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