On normal meromorphic functions

Lohwater, A. J.; Pommerenke, Ch.

doi:10.5186/aasfm.1973.550

Cited by 39 publications

(16 citation statements)

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“…The case a = 0 is due to Zalcman [28]. A similar result for normal functions had been proved earlier by Lohwater and Pornrnerenke [16]. If we assume that the zeros of the functions in F have multiplicity at least m, then the conclusion of Lemma 1 holds for -1 < cr < m. This last result was proved by Chen and Gu [8,Theorem 21. In the papers cited, the condition f #(z) 5 f #(0) = 1 is usually not mentioned, but it follows immediately from the proof.…”

Section: Rescaling Lemmassupporting

confidence: 55%

Covering properties of derivatives of meromorphic functions

Bergweiler

2001

Complex Variables, Theory and Application: An International Jou

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Section: Rescaling Lemmassupporting

confidence: 55%

Covering properties of derivatives of meromorphic functions

Bergweiler

2001

Complex Variables, Theory and Application: An International Jou

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“…Suppose, on the contrary, that f £ N. Then, by [11,Theorem 1], there are a sequence (a n ) of points in Δ and a sequence (p n ) of positive real numbers with Pn/( 1 -l 0 n|) -> 0 such that the sequence (g n (t)) = (/(a n + p n t)) of functions converges uniformly on each compact subset of C to a non-constant meromorphic function go(t). Thus…”

mentioning

confidence: 99%

On Subspaces and Subsets of Bmoa and Ubc

Aulaskari¹,

Xiao²,

Zhao³

1995

Analysis

206

133

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Some subspaces and subsets of BMOA and UBC are defined by using similar area integral conditions to the definitions of BMOA and UBC. These subspaces and subsets axe shown to be separate and included in each other. AMS Classification: 30D50 Δ Q,,ο = {/:/ analytic in Δ and Jim Jf Δ Qjf = ^ / : / meromorphic in Δ and sup J J (/*(z)) 2 g r {z,a) dx dy < oo j> and Q* 0 = I / : / meromorphic in Δ and ^lim J J ( / * ( z ) ) 2 < / p ( z , a ) dx dy = 0 Typeset by .AjuS-TfeX Brought to you by | University of Pennsylvania Authenticated Download Date | 6/14/15 4:46 PMWe can choose a pseudohyperbolic disk U((io, . <) such that U(ao,s) Π Δ \ U(0,r) φ 0 and «! 6 U(a 0 ,s) η Δ \ U(0,r). ThenBy (2.1) we find a constant C depending only on do, s and s', s < s', such that the following inequality holds IJ \1'{ζ)\ 2 9 '(ζ,α 0 )άχά ν <ϋ> j j \f'(z)\ 2 g"(z,a l )dxdy < C" < 00. (2.5) Δ\('(α0,ν) Δ\(/(α0,»') At the same time, ff \f'(z)\ 2 gP(z,a 0 )dxdy< max \f'(z)\ 2 [[ (log τ-^-λ' dxdy < 00. (2.6) J J Ι6ί/(α",5') J J \z-a 0 \' Brought to you by | University of Pennsylvania Authenticated Download Date | 6/14/15 4:46 PM 104 Aulaskari et al Combining (2.5) and (2.6) we get J J\f'(z)\ 2 g p (z,ao)dxdy < oo, Δ which contradicts (2.4). Case 2. Next suppose that, for all α € U(0, r), J J \f'(z)\ 2 g"(z,a)dxdy

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“…To prove our results, we need some preliminaries. The next is the well-known Lohwater-Pommerenke's theorem [6]. Lemma 2.1.…”

Section: Lemmasmentioning

confidence: 92%