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

Shahar Nevo¹

¹ Department of Mathematics Bar-Ilan University 52900 Ramat-Gan, Israel

Annales Polonici Mathematici, Tome 85 (2005) no. 3, pp. 251-260

Résumé

We prove the existence of sequences $\{\varrho_n\}_{n=1}^\infty$, $\varrho_n\to 0^+$, and $\{z_n\}_{n=1}^\infty$, $|z_n|= {1}/{2}$, such that for every $\alpha \in\mathbb R$ and for every meromorphic function $G(z)$ on $\mathbb C$, there exists a meromorphic function $F(z)=F_{G,\alpha}(z)$ on $\mathbb C$ such that $\varrho_n^\alpha F(nz_n+n\varrho_n\zeta)$ %%\overset \chi\Rightarrow converges to $G(\zeta)$ uniformly on compact subsets of $\mathbb C$ 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 $m\ge 1$ and on which an extension of Zalcman's Lemma holds.

Détail

DOI : 10.4064/ap85-3-6

Keywords: prove existence sequences varrho infty varrho infty every alpha mathbb every meromorphic function mathbb there exists meromorphic function alpha mathbb varrho alpha varrho zeta overset chi rightarrow converges zeta uniformly compact subsets mathbb spherical metric result construct family functions meromorphic unit disk m normal which extension zalcmans lemma holds