Summary: Let \(H(\mathbb U)\) be the space of analytic functions in the unit disk \(\mathbb U\). For the integral operator \(A^{\phi,\varphi}_{\alpha,\beta,\gamma}: {\mathcal K}\to H(\mathbb U)\) with \({\mathcal K}\subset H(\mathbb U)\), defined by \[ A^{\phi,\varphi}_{\alpha,\beta,\gamma}[f](z)=\left[\frac{\beta+\gamma}{z^\gamma \phi(z)}\int^z_0 f^\alpha(t)\varphi(t)t^{\delta-1}\text{d}t\right]^{1/\beta}, \] where \(\alpha,\beta,\gamma,\delta\in \mathbb C\) and \(\phi,\varphi\in H(\mathbb U)\), we determine sufficient conditions on \(g_1,g_2,\alpha,\beta\) and \(\gamma\), such that \[ z\varphi(z)\left[\frac{g_1(z)}{z}\right]^\alpha\prec z\varphi(z)\left[\frac{f(z)}{z}\right]^\alpha\prec z\varphi(z)\left[\frac{g_2(z)}{z}\right]^\alpha \] implies \[ z\phi(z)\left[\frac{A^{\phi,\varphi}_{\alpha,\beta,\gamma}[g_1](z)}z\right]^\beta\prec z\phi(z)\left[\frac{A^{\phi,\varphi}_{\alpha,\beta,\gamma}[f](z)}z\right]^\beta\prec z\phi(z)\left[\frac{A^{\phi,\varphi}_{\alpha,\beta,\gamma}[g_2](z)}z\right]^\beta. \] The symbol “\(\prec\)” stands for subordination, and we call such a kind of result a sandwich-type theorem. In addition, \(z\phi(z)\left[\frac{A^{\phi,\varphi}_{\alpha,\beta,\gamma}[g_1](z)}z\right]^\beta\) is the largest function and \(z\phi(z)\left[\frac{A^{\phi,\varphi}_{\alpha,\beta,\gamma}[g_2](z)}z\right]^\beta\) 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 \(\phi\) and \(\varphi\) that also generalizes classic results from the theory of differential subordination and superordination.