Gradient estimates for f-exponentially harmonic functions on complete Riemannian manifolds

For smooth metric measure spaces (M,g,e-fdvol), the gradient estimates of positive solutions to the f-exponentially harmonic functions was considered by using the maximum principle. Then a Liouville type theorem was obtained when the Bakry-Emery Ricci tensor was nonnegtive and the sectional curvature was bounded by a negative constant. This generalizes a result in Ref.Wu J, Ruan Q, Yang Y H. Gradient estimates for exponentially harmonic functions on complete Riemannian manifolds. Manuscripta Mathematica, 2014, 143(3-4): 483-489, which is covered in the case where f is a constant.