Subordination-preserving integral operators

Let β and l be complex numbers and let H be the space of functions regular in the unit disc. Subordination of functions f,g, e H is denoted by f<g. Let K⊆H and let the operator A: K→H be defined by F=A(f), where $$F(z) = {\left( {\frac{1}{{{z^\delta }}}\int_0^z {{f^\beta }(t) {t^{\delta - 1}} dt} } \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle \beta $}}}}.$$ The authors determine conditions under which $$f < g \Rightarrow A(f) < A(g)$$ and they present some applications of this result.