Eigenvalue comparison theorems for the Witten-Laplacian and the weighted $p$-Laplacian on complete manifolds with a modified Ricci curvature bounded from below

In this paper, for complete manifolds with a modified Ricci curvature bounded from below, we can successfully set up Cheng-type eigenvalue comparison theorems for the first Dirichlet eigenvalues of the Witten-Laplacian and the weighted $p$-Laplacian on geodesic balls of these manifolds.