Let f: R → R and g: R → R be two functions defined by f(x) = _ e (...

$f\left(x\right) = \log_{e}\left({x}^{2} + 1\right) - {e}^{- x} + 1 ⇒ {f}^{'}\left(x\right) = \dfrac{2x}{{x}^{2} + 1} + {e}^{- x} > 0, \forall x \in R$

So $f\left(x\right)$ increasing

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

So $g\left(x\right)$ decreasing

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

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

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

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