Maximal \(K_n\) domain of a family of rational functions
Let \(K_n(D)\) denote a class of functions \(F(z)\) analytic and univalent on a domain \(D\) for which the \(n\)th devided difference \([F(z); z_0,z_1,\dots, z_n]\), does not vanish, provided \(z_k\neq z_l\), \(k\neq l\), \(k,l= 0,1,\dots, n\). The authors are concerned with a family of functions of the form \[ F(z)= \sum^p_{k=1}\,\sum^{m_k}_{s=1} A_{k,s}(z- a_k)^{-s}, \] where constants \(A_{k,s}\), \(a_k\) satisfy certain conditions. They determine, among others, the largest disk \(\Delta_R\) centered at the origin on which \(F(z)\in K_n(\Delta_R)\).
scientific article; zbMATH DE number 2121457