  Let $(M,\theta)$ be a compact strictly pseudoconvex pseudohermitian manifold
which is CR embedded into a complex space. In an earlier paper, Lin and the
authors gave several sharp upper bounds for the first positive eigenvalue
$\lambda_1$ of the Kohn-Laplacian $\Box_b$ on $(M,\theta)$. In the present
paper, we give a sharp upper bound for $\lambda_1$, generalizing and extending
some previous results. As a corollary, we obtain a Reilly-type estimate when
$M$ is embedded into the standard sphere. In another direction, using a
Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula
for the Webster scalar curvature, we give a lower bound for $\lambda_1$ when
the pseudohermitian structure $\theta$ is volume-normalized.
