Upper bounds on the first eigenvalue for a diffusion operator via Bakry–Émery Ricci curvature

Let L = Δ − ∇ φ ⋅ ∇ be a symmetric diffusion operator with an invariant measure d μ = e − φ d x on a complete Riemannian manifold. In this paper we give an upper bound estimate on the first eigenvalue of the diffusion operator L on the complete manifold with the m-dimensional Bakry–Émery Ricci curvature satisfying Ric m , n ( L ) ⩾ − ( n − 1 ) , and therefore generalize a Chengʹs result on the Laplacian (S.-Y. Cheng (1975) [8]) to the case of the diffusion operator.