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Abstract

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.