complex function $$\varphi=u+iv\in C_{\mathbb{C}
}^{\infty}\left(  M\right)$$ and $\gamma=\gamma_{\overline{1}}\theta^{\overline{1}}\in \Omega^{0,1}\left(
M\right)  \cap \ker \left(  \square_{b}\right)  $ such that$$\overline{\partial}_{b}\varphi=\sigma_{\overline{1}}\theta^{\overline{1}}-\gamma_{\overline{1}}\theta^{\overline{1}}\label{2018}$$ with $$\overline{\partial}_{b}(\sigma_{\overline{1}}\theta^{\overline{1}})=0.$$ Here $\square_{b}=2\left(  \overline{\partial}_{b}\overline{\partial}_{b}^{\ast}+\overline{\partial}_{b}^{\ast}\overline{\partial}_{b}\right)  $ is the Kohn-Rossi Laplacian. Then it is the natural question when we have the solvability of the inhomogeneous tangential Cauchy-Riemann equation (i.e. $\gamma=0$) $$\overline{\partial}_{b}\varphi=\sigma_{\overline{1}}\theta^{\overline{1}}.\label{2018A}$$

In this paper, we mainly focus on the existence of pseudo-Einstein contact forms as in Corollary \[c2\], Theorem \[t2\], Theorem \[t3a\] and Theorem \[t4\], an upper bound eigenvalue estimate for the CR Paneitz operator as in Theorem \[t3\], Corollary \[c4\] and its applications to the uniformization theorem for Sasakian space form as in Corollary \[c4\], Corollary \[c5\] and Corollary \[c6\] in a closed strictly pseudoconvex CR $3$-manifold.

We first state one of the main theorems as follows:

\[t1\]If $(M,J,\theta)$ is an embeddable closed strictly pseudoconvex CR $3$-manifold with $c_{1}(T_{1,0}M)=0$.  Then

\(i) $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form if and only if $f$ satisfies the third-order partial differential equation$$P_{1}f=i\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)  .\label{7}$$ Here $P_{1}$ is a third-order CR-pluriharmonic operator $$P_{1}f=f_{\overline{1}11}+iA_{11}f_{\overline{1}}.$$ (ii) In particular, $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for a CR-pluriharmonic function $f$ if and only if the equality holds $$\left(  A_{11}\gamma_{\overline{1}}-\gamma_{1,0}\right)  =0.\label{13}$$

In [@le], Lempert showed that a closed 