Let H(U) be the space of analytic functions in the unit disk U. For the integral operator Aα,β,γϕ,φ:k→H(U), with K⊂H(U), defined by Aα,β,γϕ,φ[f](z)=[β+γzγϕ(z)∫0zfα(t)φ(t)tδ−1dt]1/β, where α,β,γ,δ∈ℂ and ϕ,φ∈H(U), we will determine sufficient conditions on g1, g2, α, β and γ, such that zφ(z)[g1(z)z]α≻zφ(z)[f(z)z]α≻zφ(z)[g2(z)z]α implies zϕ(z)[Aα,β,γϕ,φ[g1](z)z]β≺zϕ(z)[Aα,β,γϕ,φ[f](z)z]β≺zϕ(z)[Aα,β,γϕ,φ[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)[Aα,β,γϕ,φ[g1](z)z]β is the largest function and zϕ(z)[Aα,β,γϕ,φ[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.