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

Bergweiler, Walter; Erëmenko, Alexandre

doi:10.4171/rmi/176

Cited by 331 publications

(267 citation statements)

References 25 publications

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“…However, we can actually get by with much less. In fact, Bergweiler and Eremenko [6] have proved that a function of finite order which satisfies f f = 1 must be constant. Since the limit function g may be taken to have finite order, we see that the above proof establishes that the condition f f = 1 also implies normality.…”

Section: Examples (Continued)mentioning

confidence: 99%

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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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Section: Examples (Continued)mentioning

confidence: 99%

“…The results discussed above concerning the condition f f = 1 are due, independently and simultaneously, to Bergweiler and Eremenko [6], Chen and Fang [12], and Zalcman [71].…”

Section: Examples (Continued)mentioning

confidence: 99%

Normal families: New perspectives

Zalcman

1998

Bull. Amer. Math. Soc.

305

124

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“…[5] Multiplicities In particular Proposition 1.10 shows that $(k, k + 1, oo, C) contains no transcendental function of finite order. It is not clear whether Proposition 1.10 holds without the hypothesis t h a t / has finite order, the proof in [17] relying on a result from [2] concerning critical and asymptotic values. Indeed, it was conjectured in [1] that if/ is transcendental and meromorphic in the plane with / ' -1 zero-free then there exists a sequence z n -*• oo with / (z n ) = 0 and / '(z n ) -*• oo.…”

Section: Theorem 14 Let K M N E N and Let D Be A Non-empty Plane mentioning

confidence: 99%

“…Let i e N and let f be transcendental and meromorphic in the plane. Then, as r -» oo, 38 Walter Bergweiler and J. K. Langley [2] A normal families analogue for functions g meromorphic in a plane domain D was established in [7] (see also [15,22]). Theorems 1.1 and 1.3 together provide an illustration of the Bloch principle, that (most) properties which suffice to make constant a function meromorphic in the plane, render the corresponding family on a plane domain normal [22].…”

Section: With S(rf) Denoting Any Term Which Is O(t(rf))mentioning

confidence: 99%

Multiplicities in Hayman's alternative

Bergwelier¹,

Langley²

2005

J. Aust. Math. Soc.

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In 1959 Hayman proved an inequality from which it follows that if/ is transcendental and meromorphic in the plane then either/ takes every finite complex value infinitely often or each derivative / ( k ) , k > 1, takes every finite non-zero value infinitely often. We investigate the extent to which these values may be ramified, and we establish a generalization of Hayman's inequality in which multiplicities are not taken into account.2000 Mathematics subject classification: primary 3OD35. IntroductionOur starting point is the following result of Hayman [8,9]. THEOREM 1.1. Let k e H and let a, b e C, b ^ 0. Let f be nonconstant and meromorphic in the plane. Then at least one of f -a and f ( A r ) -b has at least one zero, and infinitely many iff is transcendental.Hayman's proof of Theorem 1.1 is based on Nevanlinna theory, the result deduced from the following inequality. Here the notation is that of [9], with S(r,f) denoting any term which is o (T(r,f)) as r -> oo, possibly outside a set of finite Lebesgue measure. (1) r(rJ)

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“…This is again standard: for the terminology see [1,42]. Assuming without loss of generality that G −1 has a direct transcendental singularity over β ∈ C \ {α 1 , α 2 } gives a small positive δ, a component U of the set {z ∈ C : |G(z) − β| < δ}, and a continuous, subharmonic, non-constant function u of finite order in the plane which satisfies u(z) = log(δ/|G(z) − β|) on U and vanishes outside U.…”

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confidence: 99%