Generalizations of Hadamard products of certain meromorphic functions (Q2717116)

Let \(f(z)= {1\over z}+\sum^\infty_{k=1} a_kz^k\) belong to \(\sum^*_p (\alpha,\beta)\), \(\alpha\in [0,1)\), \(\beta\in [0,1]\) if and only if \(f\) is analytic in \(D=\{z\mid 0<|z|<1\}\) and NEWLINE\[NEWLINE\left|{zf'(z)\over f(z)}+1 \right|<\beta\left |{zf'(z) \over f(z)}+2 \alpha-1 \right |, \;z\in D.NEWLINE\]NEWLINE The authors consider combinations NEWLINE\[NEWLINE{1\over z}+ \sum^\infty_{k=1} \prod^m_{i=1} (a_{k,k})^{1/p_i} z^k,\quad \text{resp.}\quad {1 \over z}+\sum^\infty_{k=1} \sum^m_{i=1} (a_{k,i})^q z^kNEWLINE\]NEWLINE of functions \(f_i(z)= {1\over z}+\sum^\infty_{k=1} a_{k,i}z^k \in\sum^*_p (\alpha_i,\beta)\) for \(i=1, \dots,m\) and determine classes \(\sum^*_p (\gamma, \beta)\) to which these combinations belong.