�ber ein Problem von Hayman

Mues, Erwin

doi:10.1007/bf01182271

Cited by 134 publications

(57 citation statements)

References 4 publications

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“…J. Clunie [5] proved this for the case that f is entire and n = 1. Later E. Mues [19,Satz 3] settled the case that f is meromorphic and n = 2 and W. Hennekemper [16] extended Clunie's result to functions which have few poles in some sense.…”

Section: Corollarymentioning

confidence: 99%

On the singularities of the inverse to a meromorphic function of finite order

Bergweiler¹,

Erëmenko²

1995

Rev. Mat. Iberoam.

331

257

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Our main result implies the following theorem: Let f be a transcendental meromorphic function in the complex plane. If f has finite order ρ, then every asymptotic value of f , except at most 2ρ of them, is a limit point of critical values of f .We give several applications of this theorem. For example we prove that if f is a transcendental meromorphic function then f f n with n ≥ 1 takes every finite non-zero value infinitely often. This proves a conjecture of Hayman. The proof makes use of the iteration theory of meromorphic functions.

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Section: Corollarymentioning

confidence: 99%

On the singularities of the inverse to a meromorphic function of finite order

Bergweiler¹,

Erëmenko²

1995

Rev. Mat. Iberoam.

331

257

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“…This conjecture, following partial results by Clunie [5], Mues [10] and Hayman [7], was finally confirmed by Bergweiler and Eremenko [2], Chen and Fang [4], independently.…”

Section: Introduction and Resultsmentioning

confidence: 66%

Normal Family of Meromorphic Functions

Wang¹

2014

Bulletin of the Korean Mathematical Society

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“…This condition is known to force an entire function (Hayman [27] for n ≥ 2, Clunie [16] for n = 1) or a meromorphic function on C (Hayman [27] for n ≥ 3, Mues [43] for n = 2) to be constant. The corresponding normality results are due to Yang and Zhang [66] (for analytic functions, n ≥ 2) and [67] (for meromorphic functions, n ≥ 5), Gu [32] (for meromorphic functions, n = 3, 4), Oshkin [45] (for analytic functions, n = 1; cf.…”

Section: Examples (Continued)mentioning

confidence: 99%

Normal families: New perspectives

Zalcman

1998

Bull. Amer. Math. Soc.

305

124

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Abstract. This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol'dberg's Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch's Principle and provide simple solutions to some problems of Hayman connected with this principle.Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch's Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families 1 (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report.One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale.

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�ber ein Problem von Hayman

Cited by 134 publications

References 4 publications

On the singularities of the inverse to a meromorphic function of finite order

On the singularities of the inverse to a meromorphic function of finite order

Normal Family of Meromorphic Functions

Normal families: New perspectives

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