A refined Schwarz lemma for $V$-harmonic maps

In this note, we proved a refined Schwarz lemmas for $V$-harmonic maps. Specifically, we prove that if $u$ is a $V$-harmonic map of generalized dilatation of order $\beta$ from a complete Riemannian manifold with Bakry–Émery Ricci curvature bounded below by a constant $-A$ to a Riemannian manifold with sectional curvature bounded above by a negative constant $-B$, then $u^{\ast }h\le \frac{A{\beta }^2}{B}g$.