$$R>f\left(  s\right)  :=-2\left[  \left(  C_{0}b\right)  -\frac{\left(
2C_{0}b+C_{1}d\right)  +\left(  2C_{0}a+C_{1}c\right)  s}{1+s^{2}}\right]
.\label{41b}$$ Since $X$ is arbitrary, this inequality holds for all $s\in \mathbb{R}$. We have $$\begin{array}
[c]{ccl}f(s) & \leq & \max_{s\in \mathbb{R}}f(s)\\
& = & f(s_{0})\\
& = & 2\sqrt{\left(  C_{0}a+C_{1}\frac{c}{2}\right)  ^{2}+\left(  C_{0}b+C_{1}\frac{d}{2}\right)  ^{2}}+C_{1}d\\
& \leq & 2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)
\end{array}
\label{41a}$$ where $$s_{0}=\frac{-\left(  C_{0}b+C_{1}\frac{d}{2}\right)  +\sqrt{\left(
C_{0}a+C_{1}\frac{c}{2}\right)  ^{2}+\left(  C_{0}b+C_{1}\frac{d}{2}\right)
^{2}}}{\left(  C_{0}a+C_{1}\frac{c}{2}\right)  },$$ for $C_{0}a+C_{1}\frac{c}{2}\neq0$, is a critical number of $f$. Thus $$R(x)>2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)  (x)$$ for all $x\in M$. As for the case of $C_{0}a+C_{1}\frac{c}{2}=0$, the same deduction could be applied. Therefore, $\left(  C_{0},C_{1}\right)
$-positivity is equivalent to $$R>2\left(  C_{0}|A_{11}|+C_{1}\left \vert A_{11,\overline{1}}\right \vert
\right)  .$$

([@l]) (i) A contact form $\theta$ on a closed strictly pseudoconvex CR $(2n+1)$-manifold $(M,\theta)$ is said to be pseudo-Einstein for $n\geq2$ if the pseudohermitian Ricci tensor $R_{\alpha \overline{\beta}}$ is proportional to the Levi form $h_{\alpha \overline{\beta}}$, i.e., $$\ R_{\alpha \overline{\beta}}=\frac{R}{n}h_{\alpha \overline{\beta}},$$ where $R=h^{\alpha \overline{\beta}}R_{\alpha \overline{\beta}}$ is the Tanaka-Webster scalar curvature of $(J,\theta).$

\(ii) (Lemma \[l1\]) Note that any contact form on a closed strictly pseudoconvex $3$-manifold is actually pseudo-Einstein (since the pseudohermitian Ricci tensor has only one component $R_{1\overline{1}}$). Then we define a contact form $\theta$ on a closed strictly 