Universal sequences for Zalcman's Lemma and Q~ m-normality

We prove the existence of sequences {ρn}∞ n=1, ρn → 0 +, |zn| = 1 2 , such that for every α ∈ R and for every meromorphic function G(z) on C, there exists a corresponding meromorphic function F(z) = FG,α(z) on C, such that ρ α nF(nzn + nρnζ) χ⇒ G(ζ) on C. As a result, we construct a family of functions meromorphic on the unit disk ∆ that is not Qm-normal for every m ≥ 1 and on which an extension of Zalcman's Lemma holds.