On certain subclasses of meromorphic functions (Q1920702)

Let the class \(\Sigma\) consist of all functions of the form \[ f(z)= 1/z+ \sum^\infty_{n= 0} a_n z^n,\quad a_n\geq 0,\quad z\in E_0= \{z: 0< |z|< 1\} \] and \(D^n f(z)= {1\over z(1+ z)^{n+ 1}}* f(z)\). Denote by \({\mathcal M}_n(A, B)\) the class of all \(f\) such that \[ \text{Re}\Biggl\{ {D^{n+ 1} f(z)\over D^n f(z)}- 2\Biggr\}< {n(1+ B)+ 1+ A\over (n+ 1)(1+ B)}, \] where \(- 1\leq A< B\), \(B> 0\). Denote by \({\mathcal P}_n(A, B)\) the subclass of \({\mathcal M}_n(A, B)\) that consists of functions belonging to \(\Sigma\). In the paper, the author proves that functions in the class \({\mathcal M}_n(A, B)\) are univalent in \(E_0\) and he considers certain integral operators and proves that \({\mathcal M}_n(A, B)\) and \({\mathcal P}_n(A, B)\) are closed under these operators. He also proves a distortion theorem.