Let $f:R \to R$ and $g:R \to R$ be two functions defined by $f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$ and $g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}$. Then, for which of the following range of $\alpha$, the inequality $f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha -{5 \over 3}} \right)} \right)$ holds ?