Parametric representation and description for the root sets of weighted classes of holomorphic functions in the disc
Let \(D = \{z:|z|<1\}\) be the unit disk in the complex plane and let \(H(D)\) be the set of all holomorphic functions in \(D\). Denote by \(N_{\omega}^p\) the function class \[ N_{\omega}^p = \left\{f\in H(D): \|T(f)\|_{L^p(\omega)} =\left( \int_0^1\omega (1-r)T(f,r)^p dr\right)^{1/p} < +\infty\right\}, \] where \(T(f,r)\) is the Nevanlinna characteristic of a function \(f\) and \(\omega\) is a positive function in \(L^1(0,1)\). For \(p=1\), \(\omega = t^{\alpha}\), \(\alpha > -1\), the class \(N_{\alpha} = N_{\alpha}^1\) was introduced by Nevanlinna. The author studies the case in which the weight function has a more general form and \(p<\infty\). More precisely, it is assumed that \(\omega\in S\), where \(S\) is a set of positive measurable functions on \((0,1)\) for which there exist numbers \(m_{\omega}\), \(M_{\omega}\), and \(q_{\omega}\), with \(m_{\omega}\), \(q_{\omega}\in (0,1)\), such that \[ m_{\omega}\leq\frac{\omega(\lambda r)}{\omega (r)}\leq M_{\omega},\quad r\in (0,1),\;\lambda \in [q_{\omega}, 1]. \] The author finds a complete characterization for the root sets and constructs a parametric representation for the class \(N_{\omega}^p\). As was expected, the root characteristic of the class \(N_{\omega}^p\) depends essentially on \(p\) and \(\omega\).
scientific article; zbMATH DE number 1439057