I want to prove the following inequality for $x \in (0,1)$: \begin{eqnarray*} \Gamma(x+1)>\frac{\frac{(900\gamma^2+73\pi^2)x}{900\gamma}+\frac{73}{100}}{\frac{\pi^2 x}{9\gamma}+1}\quad \tag{1} \end{eqnarray*}
Let \begin{align*} f(x) &:=\log \left(\Gamma(x+1) \left(\frac{\pi^2 x}{9\gamma}+1\right) \right) \\ g(x) &:=\log \left( \frac{(900\gamma^2+73\pi^2)x}{900\gamma}+\frac{73}{100} \right). \end{align*}
and define
\begin{eqnarray*} w(x) := \frac{f(x)}{g(x)} , \end{eqnarray*}
For $x \in (0, 1)$ $g(x)>0$ holds, but $g$ and $f(x)$ are not equal to zero! What it need to get it valid for this condition (theorem)?