On certain classes of meromorphically \(p\)-valent starlike functions
Let \(\sum_ p\) denote the class of functions \(f(z) = \sum^ \infty_{k = - p} a_{k + p - 1} z^ k\) with \((a_{-1}=1)\) which are analytic for \(0<| z |<1\) and define \(D^{n+p-1} f(z) = z^{-p} (1-z)^{-n- p}*f(z)\) where \(*\) indicates the Hadamard product. The authors consider the classes \(M_{n+p-1} (\alpha)\) for which \[ \text{Re}\{z(D^{n+p- 1}f(z))'/ D^{n+p-1} f(z)\}<-\alpha \] in this annulus where \(0 \leq \alpha<p\). They then prove that these classes are nested, \(M_{n + p} (\alpha) \subset M_{n+p-1} (\alpha)\).
scientific article; zbMATH DE number 612154
scientific article