Properties of some families of meromorphic \(p\)-valent functions (Q2709957)

The author introduces another two highly specialized subclasses of meromorphic \(p\)-valent functions. Define functions \(\varphi_p (a,c;z) = z^{-p} + \sum_{k=1}^\infty (a)_k/(c)_k z^{k-p}\) where \(c \neq 0, -1, -2, \dots\) and \((x)_k = x(x+1)\dots (x+k-1)\). Then for \(f\) of the form \(f(z) = z^{-p} + \sum_{k=1}^\infty a_{k-p} z^{k-p}\) the linear operator \(L_p(a,c)f(z) = \varphi_p (a,c;z)\ast f(z)\) (where \(\ast\) denotes the Hadamard product) is defined. Then the class \(S_{a,c}(A,B,\alpha)\) is defined to be of those \(f\) with \(|[z^{p+1} (L_p(a,c)f(z))' + p]/[Bz^{p+1}(L_p(a,c)f(z))' +Ap|< \alpha\). A second starlike class is defined similarly. A number of properties of these classes are proved.