pseudo-Einstein contact form. Moreover, if the torsion is vanishing, then $(M,J,\theta)$ is the Sasakian space form.

(proof of Theorem \[t3a\] and Corollary \[c5\]) As before $$\begin{array}
[c]{ccl}\int_{M}Qu^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot}d\mu & = &
\int_{M}Q^{\bot}u^{\bot}d\mu+\int_{M}\left(  P_{0}u^{\bot}\right)  u^{\bot
}d\mu \\
& \geq & \Lambda \int_{M}(u^{\bot})^{2}d\mu \text{ }\\
& \geq & 0
\end{array}$$ if $$Q^{\bot}=0.\label{2018G}$$ It follows from (\[2018F\]) that $$0\geq \int_{M}\left(  R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right)  \left(
\gamma,\gamma \right)  d\mu+\int_{M}\left \vert \gamma_{1,1}\right \vert ^{2}d\mu.\label{2018BBB}$$ if (\[2018G\]) holds. Hence$$\gamma=0\text{ }$$ if the pseudohermitian curavture is $\left(  \frac{1}{2},\frac{1}{2}\right)
$*-positive.* It follows from Theorem \[t2\] that $M$ admits a globally defined pseudo-Einstein contact form $\widetilde{\theta}=e^{\frac{\left(  f+2u\right)  }{3}}\theta$.  

Furthermore, if he CR  $Q$-curvature is pluriharmonic (i.e. $Q^{\bot}=0),$ then $$u^{\bot}=0$$ and by (\[0\]) $$W_{1}=0.$$ Hence $\theta$ is also a globally defined pseudo-Einstein contact form.

Now if $(M,J,\theta)$ is spherical and pseudo-Einstein, we have $$W_{1}=R,_{1}-iA_{11,\overline{1}}=0$$ and $$iR_{,11}=3RA_{11}+6iA_{11,0}-4A_{11,\overline{1}1}.$$

By cancelling $R_{,11}$, one derives$$3RA_{11}+6iA_{11,0}-3A_{11,\overline{1}1}=0.$$

On the other hand, it follows from the commutation relation ([@l]) that$$A_{11,1\overline{1}}-A_{11,\overline{1}1}=iA_{11,0}+2RA_{11},$$

we obtain$$-3RA_{11}+2A_{11,1\overline{1}}-3A_{11,\overline{1}1}=0$$ and then$$-2{\displaystyle \int_{M}}
\left \vert A_{11,1}\right \vert ^{2}d\mu+3{\displaystyle \int_{M}}
\left \vert A_{11,\overline{1}}\right \vert ^{2}d\mu=3{\displaystyle \int_{M}}
R\left \vert A_{11}\right \vert ^{2}d\mu.$$ Moreover, if $R$ is positive constant, then $$\text{\ }A_{11,\overline{1}}=0$$ and $$-2{\displaystyle \int_{M}}
\left \vert A_{11,1}\right \vert ^{2}d\mu=3{\displaystyle \int}
R\left \vert A_{11}\right \vert ^{2}$$ which implies $$A_{11}=0.$$ It 