  In this note, we mainly focus on the existence of pseudo-Einstein contact
forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its
applications to the uniformization theorem for Sasakian space form in an
embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence
of pseudo-Einstein contact form is confirmed if the CR 3-manifold is Sasakian.
Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitz
operator and obtain the CR uniformization theorem for a class of CR
3-manifolds. At the end, under the positivity assumption of the pseudohermitian
curvature, we derive the existence theorem for pseudo-Einstein contact forms
and uniformization theorems in a closed strictly pseudoconvex CR 3-manifold of
nonnegative CR Paneitz operator with kernel consisting of the CR-pluriharmonic
functions and the CR Q-curvature is CR-pluriharmonic.
