Let $a> 1, \alpha<0$ and $\beta>0$. How to prove that the function: $$f(x)=\frac{\Gamma(a)-\Gamma(a,\alpha \log(\beta x))}{(\alpha\log(\beta x))^a}\cdot \frac{(\alpha\log(x))^a}{\Gamma(a)-\Gamma(a,\alpha \log(x))} \quad 0<x<1,$$ is decreasing for $a<1$ and increasing for $a>1$.

This question is motivated by the following inequality after drawing the graph for some values with wolfram. I tried the approach in the above link but doesn't work in this case. Also the sign of derivative is more delicate.

Maybe one would have a simple idea, but any suggestion would be helpful.