Universal sequences for Zalcman’s Lemma and $Q_m$-normality

Shahar Nevo. "Universal sequences for Zalcman’s Lemma and $Q_m$-normality." Annales Polonici Mathematici 85.3 (2005): 251-260. <http://eudml.org/doc/280358>.

@article{ShaharNevo2005,
abstract = {We prove the existence of sequences $\{ϱₙ\}_\{n=1\}^∞$, ϱₙ → 0⁺, and $\{zₙ\}_\{n=1\}^∞$, |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function $F(z) = F_\{G,α\}(z)$ on ℂ such that $ϱₙ^α F(nzₙ + nϱₙζ)$ converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is $Q_m$-normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.},
author = {Shahar Nevo},
journal = {Annales Polonici Mathematici},
keywords = {-normal family; Zalcman's Lemma},
language = {eng},
number = {3},
pages = {251-260},
title = {Universal sequences for Zalcman’s Lemma and $Q_m$-normality},
url = {http://eudml.org/doc/280358},
volume = {85},
year = {2005},
}

TY - JOUR
AU - Shahar Nevo
TI - Universal sequences for Zalcman’s Lemma and $Q_m$-normality
JO - Annales Polonici Mathematici
PY - 2005
VL - 85
IS - 3
SP - 251
EP - 260
AB - We prove the existence of sequences ${ϱₙ}_{n=1}^∞$, ϱₙ → 0⁺, and ${zₙ}_{n=1}^∞$, |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function $F(z) = F_{G,α}(z)$ on ℂ such that $ϱₙ^α F(nzₙ + nϱₙζ)$ converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is $Q_m$-normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.
LA - eng
KW - -normal family; Zalcman's Lemma
UR - http://eudml.org/doc/280358
ER -