An inclusion operator in spaces of analytic functions (Q1066307)

A sample result: Denote by \(A^ p_{\alpha}\) the Bergman-Dzhrbashyan class of analytic functions on the unit disc \({\mathbb{D}}\) \((-1<\alpha <\infty\), \(0<p<\infty)\). For \(a\in {\mathbb{D}}\) and \(0<\eta <1\) define \[ S_{\eta}(a)=\{z\in {\mathbb{D}}:| a-z| <\eta (1-| a|)\}. \] Fix \(q\geq p\). Then a sequence \(\{z_ k\}\) of points of \({\mathbb{D}}\) is weakly separated if and only if \[ \sum_{k}\max_{t\in S_{\eta}(z_ k)}| f(t)| (1-| z_ k|^ 2)^{(2+\alpha)q/p}\quad <\infty \] for every f in \(A^ p_{\alpha}\).