Let H(U) be the space of analytic functions in the unit disk U. For the integral operator AΦ,φα,β,ν : K → H(U), with K ⊂H(U), defined by
AΦ,φα,β,ν[f](z) =[β+ν/zνΦ(z)∫z0f (t)φ(t)tδ−1dt]1/ β,
where α, β, ν, δ ∈C and Φ, φ ∈ H(U), we will determine sufficient conditions on g1, g2, α, β and ν, such that
zφ(z)[g1(z)/z]α< zφ(z)[ f(z)/z]α< zφ(z)[g2(z)/z]α
implies
zΦ(z)[AΦ,φα,β,ν[g1](z)/z]β< zΦ(z)[AΦ,φα,β,ν[f](z)/z]β< zΦ(z)[AΦ,φα,β,ν[g2](z)/z]β
.
The symbol “<” stands for subordination, and we call such a kind of result a sandwich-type theorem. In addition, zΦ(z)[AΦ,φα,β,ν[[g1](z)/z]β is the largest function and zΦ(z)[AΦ,φα,β,ν[g2](z)/z]β the smallest function so that the left-hand side, respectively the right-hand side of the above implications hold, for all f functions satisfying the assumption. We give a particular case of the main result obtained for appropriate choices of functions Φ and φ that also generalizes classic results of the theory of differential subordination and superordination.
