Subordination-preserving integral operators

Abstract

Let β \beta and γ \gamma be complex numbers and let H H be the space of functions regular in the unit disc. Subordination of functions f f , g ∈ H g \in H is denoted by f ≺ g f \prec g . Let K ⊂ H K \subset H and let the operator A : K → H A:K \to H be defined by F = A ( f ) F = A(f) , where \[ F ( z ) = [ 1 z γ ∫ 0 z f β ( t ) t γ − 1 d t ] 1 / β . F(z) = {\left [ {\frac {1} {{{z^\gamma }}}\int _0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right ]^{1/\beta }}. \] The authors determine conditions under which \[ f ≺ g ⇒ A ( f ) ≺ A ( g ) , f \prec g \Rightarrow A(f) \prec A(g), \] and then use this result to obtain new distortion theorems for some classes of regular functions.