Let f : R → R and g : R → R be two functions defined by f x = log e x 2 + 1 - e - x + 1 and g x…

$f\left(x\right)={\log }_e\left(x^2+1\right)-e^{-x}+1\Rightarrow f^{\text{'}}\left(x\right)=\frac{2x}{x^2+1}+e^{-x}>0,\forall x\in R$

So $f\left(x\right)$ increasing

$g\left(x\right)=e^{-x}-2e^x \Rightarrow g^{\text{'}}\left(x\right)=-\left(e^{-x}+2e^x\right)<0 \forall x\in R$

So $g\left(x\right)$ decreasing

$\Rightarrow f\left(g\left(x\right)\right)$ is decreasing

$\Rightarrow f\left(g\left(\frac{(\alpha -1{)}^2}{3}\right)\right)>f\left(g\left(\alpha -\frac{5}{3}\right)\right)$

$\Rightarrow \frac{{\left(\alpha -1\right)}^2}{3}<\alpha -\frac{5}{3}$

$\Rightarrow \left(\alpha -2\right)\left(\alpha -3\right)<0 \Rightarrow \alpha \in \left(2,3\right)$