positivity of pseudohermitian curvature and indicate the geometry and topology of strictly pseudoconvex CR $3$-manifolds. Let $C_{0},C_{1}$ be both nonnegative numbers.

We say that a strictly pseudoconvex CR $3$-manifolds is $\left(  C_{0},C_{1}\right)  $*-convex* if the pseudohermitian curvature is $\left(
C_{0},C_{1}\right)  $-positive. That is$$\begin{array}
[c]{ccl}(R-C_{0}Tor-C_{1}Tor^{\prime})(X,X) & = & Rx^{1}x^{\bar{1}}-2C_{0}\mathrm{Re}[i(A_{\bar{1}\bar{1}}x^{\bar{1}}x^{\bar{1}})]\\
&  & -2C_{1}\mathrm{Re[}(iA_{\bar{1}\bar{1},1}x^{\bar{1}})]>0,
\end{array}
\label{33}$$ $\mathrm{for\ any}\ X=x^{1}Z_{1}\in T_{1,0}(M).$

Before giving the proof of Theorem \[t3\], we explain why we introduce the notion of $\left(  C_{0},C_{1}\right)  $-convexity as follows:

\[l6\]Let $M$ be a closed strictly pseudoconvex CR $3$-manifold. For any nonnegative constant $C_{0},C_{1}$; $\left(  C_{0},C_{1}\right)  $-convexity is equivalent to the curvature-torsion pinching condition $$R(x)>2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)  (x)$$ for all $x\in M$.

\[r2\] ([@cacc]) Let $(M^{3},J,\theta)$ be a closed strictly pseudoconvex CR $3$-manifold with $$R(x)>\left \vert A_{11}(x)\right \vert$$ which is $\left(  C_{0},0\right)  $-positive. Then $M$ admits a Riemannian metric of positive scalar curvature.

Fix a point $x\in M$ and denote $$\left \{
\begin{array}
[c]{l}A_{11}(x)=a(x)+b(x)i\\
A_{11,\overline{1}}(x)=c(x)+d(x)i
\end{array}
\right.  .$$ Without loss of generality, one may consider $X=x^{1}Z_{1}=(1+si)Z_{1}$ for some $s\in \mathbb{R}$. The convexity condition $\left(  \ref{33}\right)  $ reads as $$\begin{array}
[c]{ccl}R(1+si)(1-si) & > & i[C_{0}(a-bi)(1-si)^{2}+C_{1}\left(  c-di\right)
(1-si)-conj.]\\
& = & -2\left[  \left(  C_{0}b\right)  s^{2}-\left(  2C_{0}a+C_{1}c\right)
s+\left(  -C_{1}d-C_{0}b\right)  \right]  .
\end{array}$$ i.e., 