Subordination preserving integral operators

Abstract

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, ε 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.

This work was carried out while these authors were U.S.A.-Romanian Exchange Scholars.