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

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Nevo S.

Annales Polonici Mathematici

2005

tom 85

nr 3

251-260

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Nevo S.

1 2005

Wyniki wyszukiwania

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