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

Shahar Nevo

Universal sequences for Zalcman’s Lemma and

Q_{m}

-normality

Shahar Nevo

Annales Polonici Mathematici, Tome 85 (2005), p. 251-260 / Harvested from The Polish Digital Mathematics Library

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Résumé

We prove the existence of sequences ${ϱ ₙ}_{n = 1}^{\infty}$ , ϱₙ → 0⁺, and ${z ₙ}_{n = 1}^{\infty}$ , |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 $ϱ ₙ^{α} F (n z ₙ + n ϱ ₙ ζ)$ 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.

Publié le : 2005-01-01

Zbl 1082.30023

EUDML-ID : urn:eudml:doc:280358

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Shahar Nevo. Universal sequences for Zalcman’s Lemma and $Q_m$-normality. Annales Polonici Mathematici, Tome 85 (2005) pp. 251-260. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap85-3-6/