Canonical Blaschke-type products for the Nevanlinna area class (Q1067041)

Let \({\mathcal N}\) be the Nevanlinna area class of functions f analytic in the unit disc and satisfying \[ \iint_{| z| <1}\log^+| f(z)| dxdy<\infty,\quad z=x+iy. \] For \(\{z_ n\}_{n\geq 1}\subset \{| z| <1\}\) satisfying \(\sum_{n}(1-| z_ n|)^ 2<\infty\) and \(\epsilon >0\) it is shown that the product \[ P_{\epsilon}(z_ n,z)=\prod_{n\geq 1}(1-((1-| z_ n|^ 2)/(1-\bar z_ nz))^{2+\epsilon}) \] belongs to \({\mathcal N}\) and for \(\epsilon\leq 4\), \(P_{\epsilon}(z_ n,z)\) has zeros precisely \(\{z_ n\}_{n\geq 1}.\) It is also proved that any function \(f\in {\mathcal N}\), \(f\not\equiv 0\), can be represented as \(f(z)=z^ kP_{\epsilon}(z_ n,z)g_{\epsilon}(z)\) with given \(\epsilon\in (0,4]\), where \(g_{\epsilon}(z)\in {\mathcal N}\) and \(g_{\epsilon}(z)\) does not vanish in \(\{| z| <1\}\).