Applications of Zalcman's lemma to \(Q_m\)-normal families (Q2757215)

A local version of the following lemma due to \textit{L. Zalcman} [Am. Math. Mon. 82, 813--817 (1975; Zbl 0315.30036)] is generalized.NEWLINENEWLINENEWLINEZalcman's lemma. A family \(F\) of functions meromorphic (resp. analytic) on the unit disk \(\Delta\) is not normal in the sense of Montel if there exist a sequence of points \(z_n\), \(|z_n|< r\) \((0< r< 1)\) and a sequence of functions \(f_n\in F\) as well as a sequence of numbers \(\rho_n\to 0^+\), such that \(f_n(z_n+ \rho_z\varsigma)\to g(\varsigma)\) locally \(\chi\)-uniformly (resp. locally uniformly) on \(\mathbb C\), where \(g\) is a nonconstant meromorphic (entire) function on \(\mathbb C\). Moreover, \(g(\varsigma)\) can be taken such that \(g^\#(\varsigma)\leq g^\#(0)= 1\), \(\varsigma\in \mathbb C\),w here \(g^\#(\varsigma)= {|g'(\varsigma)|\over 1+|g(\varsigma)|^2}\) is the spherical derivative, while locally \(\chi\)-uniformly means locally uniformly w.r.t. the spherical metric \(\chi\) on \(\widehat {\mathbb C}\). NEWLINENEWLINEBesides, it is also proved an extension of the above lemma to quasi-normal families, \(Q_m\)-families of finite order and general \(Q_m\)-families. We refer to the \textit{C.-T. Chuangs'} monograph [Normal families of meromorphic functions. Singapore: World Scientific (1993; Zbl 0878.30026)] for the theory of quasi-normal family and \(Q_m\)-quasi-normal families of finite order.