Subordination-preserving integral operators

Abstract:

Let $\beta$ and $\gamma$ be complex numbers and let $H$ be the space of functions regular in the unit disc. Subordination of functions $f$, $g \in H$ is denoted by $f \prec g$. Let $K \subset H$ and let the operator $A:K \to H$ be defined by $F = A(f)$, where \[ F(z) = {\left [ {\frac {1} {{{z^\gamma }}}\int _0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right ]^{1/\beta }}.\] The authors determine conditions under which \[ f \prec g \Rightarrow A(f) \prec A(g),\] and then use this result to obtain new distortion theorems for some classes of regular functions.