as in section $2$ and section $3,$ another Bôchner-type equality holds$$\int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu+\int_{M}Qud\mu+\int_{M}\left(  P_{0}u\right)  ud\mu=0.$$ With the help of the notion of $\left(  C_{0},C_{1}\right)  $*-convexity*, we have the eigenvalue estimate for the CR Paneitz operator $P_{0}$ in terms of the CR  $Q$-curvature*.*

\[t3\] Let $(M,J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold of $c_{1}(T_{1,0}M)=0$ and nonnegative CR Paneitz operator $P_{0}$ with kernel consisting of the CR-pluriharmonic functions. Assume that  the pseudohermitian curvature is $\left(  \frac{1}{2},\frac{1}{2}\right)
$-positive $$R>(|A_{11}|+\left \vert A_{11,\overline{1}}\right \vert ).$$ If $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$ is a pseudo-Einstein contact form for any CR-pluriharmonic function $f$, then one can derive the upper bound estimate for the first eigenvalue of the CR Paneitz operator $P_{0}$ $$\Lambda^{2}\int_{M}(u^{\bot})^{2}d\mu \leq \int_{M}(Q^{\bot})^{2}d\mu
\label{2018H}$$ with the decomposition $Q=Q_{\ker}+Q^{\perp}$ and $u=u_{\ker}+u^{\perp}$. Here $\Lambda$ is the positive constant as in (\[41\]).

1\. ([@h]) For a closed strictly pseudoconvex CR $3$-manifold of vanishing pseudohermitian torsion (Sasakian), we have $$\ker P_{1}=\ker P_{0}.\label{2018C}$$ Furthermore, the real ellipsoids in $\mathbf{C}^{2}$ are such that the CR Paneitz operators are nonnegative with kernel consisting of the CR-pluriharmonic functions ([@ccya]). In general for non-embeddable CR $3
$-manifolds, we only have $$\ker P_{1}\varsubsetneq \ker P_{0}.$$

2\. As in the Remark 