Shahar Nevo

On theorems of yang and schwick

Let D be a plane domain a meromorphic function on D and k a fixed positive integer. Let F be a collection of functions meromorphic on D, none of which have poles in common with φ. According to a result of Schwick (cf. Yang [9]), if each f ε F satisfies for z ε D then .F is a normal family. We give a very simple proof of this result, based on applying a suitable refinement of Zalcmans Lemma.

APPLICATONS OF ZALCMAN'S LEMMA TO Qm-NORMAL FAMILIES

A family F of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in F has a subsequence which converges locally χ-uniformly on D \E, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of quasi-normal family further in an inductive fashion. According to Chuang, a family F of meromorphic functions on a plane domain D is Qm-normal (m = 0, 1, 2, . . . ) if each sequence S of functions in F has a subsequence which converges locally χ-uniformly on the domain D \ E, where E = E(S) ⊂ D satisfies E D = ∅. (Here E (m) D is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper gives generalizations of Zalcman’s Lemma to Qm-normal families, together with applications of these generalizations – specifically, determining the degree of normality (m) of families of meromorphic functions obtained as linear combinations of functions taken from given families.

Derivatives of Meromorphic Functions with Multiple Zeros and Rational Functions

Let f be a transcendental meromorphic function on ℂ, all but finitely many of whose zeros are multiple, and let R be a rational function, R ≢ 0. Then f′ − R has infinitely many zeros.

Picard-Hayman behavior of derivatives of meromorphic functions with multiple zeros

The derivative of a transcendental meromorphic function all of whose zeros are multiple assumes every nonzero complex value infinitely often. In 1959, Hayman [3] proved the following seminal result, which has come to be known as Hayman’s Alternative. Theorem A. Let f be a transcendental meromorphic function on the complex plane C. Then either (i) f assumes each value a ∈ C infinitely often, or (ii) f (k) assumes each value b ∈ C \ {0} infinitely often for k = 1, 2, . . . . Considering the function g(z) = [f(z)− a]/b shows that it suffices to take a = 0 and b = 1 in Theorem A. Associated with Theorem A are the following companion results. Theorem B. Let f be a meromorphic function on C. If f(z) = 0 and f (z) = 1 for some fixed positive integer k and all z ∈ C, then f is constant. Theorem C. Let F be a family of meromorphic functions on a plane domain D. Suppose that for each f ∈ F , f(z) = 0 and f (z) = 1 for some fixed positive integer k and all z ∈ D. Then F is a normal family on D. Theorem B is an immediate consequence of Theorem A, which shows that no transcendental meromorphic function can satisfy f(z) = 0, f (z) = 1 for all z ∈ C. On the other hand, if f is a nonconstant rational function such that f(z) = 0 for z ∈ C, then f(∞) = 0 for each k ≥ 1, so f (k) assumes every value with the possible exception of 0 in the finite plane. Theorem C, a celebrated result of Gu [4], is related to Theorem B via Bloch’s Principle [8, p. 222]; for a very simple proof along these lines, see [8, p. 225]. In recent years, it has become clear that, in many instances, the condition f = 0 can be replaced by the assumption that all zeros of f have sufficiently high multiplicity. This announcement concerns such an extension of Theorem A. We restrict our attention to the case k = 1. Received by the editors November 23, 2005. 2000 Mathematics Subject Classification. Primary 30D35, 30D45.

Quasinormal Families of Meromorphic Functions

Let \(F\) be a quasinormal family of meromorphic functions on D, all of whose zeros are multiple, and let ϕ be a holomorphic function univalent on D. Suppose that for any f ∊ \(F\) , f′(z) ≠ ϕ′(z) for z ∊ D. Then \(F\) is quasinormal of order 1 on D. Moreover, if there exists a compact set K ⊂ D such that each f ∊ \(F\). vanishes at two distinct points of K, then \(F\) is normal on D.

Transfinite Extension to Q m-normality Theory

The theory of Qm-normal families, m ∈ℕ, was developed by P. Montel for the cases m = 0 (normal families) [5] and m = 1 (quasinormal families) [4] and later generalized by C.T. Chuang [2] for any m ≥ 0. In this paper, we extend the definition to an arbitrary ordinal number α as follows. Given E ⊂D, define the α-th derived set % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!

On a result of Singh and Singh concerning shared values and normality

E^{(\alpha)}_D

Normality Properties of the Family f(nz): n ∈ N

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Quasi-normality induced by differential inequalities

(E^{(\alpha-1)}_D)^{(1)}_D

Q α-Normal Families and Entire Functions

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