A generalization of almost Schur lemma on CR manifolds

In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds $(M,J,\theta )$ for $n\ge 2$. When the equality of almost Schur inequality holds, we derive the contact form $\theta$ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.