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Let $L=\Delta-\nabla\varphi\cdot\nabla$ be a symmetric diffusion operatorwith an invariant measure $d\mu=e^{-\varphi}dx$ on a complete Riemannianmanifold. In this paper we prove Li-Yau gradient estimates for weightedelliptic equations on the complete manifold with $|\nabla \varphi|\leq\theta$and $\infty$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by somenegative constant. Based on this, we give an upper bound on the firsteigenvalue of the diffusion operator $L$ on this kind manifold, and therebygeneralize a Cheng's result on the Laplacian case (Math. Z., 143 (1975)289-297).

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II