Let H(U) be the space of analytic functions in the unit disk U.For the integral operator Aφ,(ψ)α,β,γ:K → H(U),with K (C) H(U),defined by Aφ,(ψ)α,β,γ[f](z) =[β+γ/zγφ(z)∫z0fα(t)(ψ)(t)tδ-1 dt]1/β, where α,β,γ,δ ∈ (C) and φ,(ψ) ∈ H(U),we will determine sufficient conditions on g1,g2,α,βand γ,such thatz(ψ)(z)[Aφ(ψ)α,β,γ[g1](z)/z]β(＜)z(ψ)(z)[f(z)/z]β(＜)z(ψ)(z)[g3(z)/z]αimplieszφ(z)[Aφ,(ψ)α,β,γ[g1](z)/z]β (＜)zφ(z)[Aφ,(ψ)α,β,γ[f](z)/z]β(＜) zφ(z)[Aφ,(ψ)α,β,γ[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,(ψ)α,β,γ[g1](z)/z]β is the largest function and [Aφ,(ψ)α,β,γ[g2](z)/z]βφ(z) 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 (ψ),that also generalizes classic results of the theory of differential subordination and superordination.