Generalization properties for certain analytic functions (Q1273328)

The results obtained by \textit{H. Saitoh} (Math. Jap. 35, No. 6, 1073-1076 (1990; Zbl 0723.30009)] on a set of functions of the type \[ F(\alpha,\beta,z) =\alpha f(z)+\beta zf'(z) \] where \(f(z) = z + a_{n+1}z^{n+1} +\dots\) are analytic in the open unit disk, are generalized. The proofs of these generalizations have been made by means of the well-known method, called that ``of admissible functions'' based on differential subordinations, order 1 and 2, introduced by S. S. Miller and P. T. Mocanu in 1978 and developed later on.