Inverse problem for zeros of certain Koebe-related functions (Q1361060)

The authors consider the zero set of a Koebe type function \(K_{\lambda,\mu}^{(n)}(z)=1-\sum_{k=1}^n \mu_k K(\lambda_k z)\), where \(K\) denotes the Koebe function \(z(1-z)^{-2}\), \(\lambda_k\in{\mathbf C}\), \(|\lambda_k|=1\), \(\mu_k > 0\). The following statement is proved: If \(w_1,\dots, w_n\) is a sequence of nonzero points in the unit disk \({\mathbf D}\), then there is the unique function \(K_{\lambda, \mu}^{(n)}\) such that \(w_1,\dots, w_n\) coincides with the zero set in \({\mathbf D}\) of the function \(K_{\lambda,\mu}^{(n)}\).