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 $a>1$. $\Gamma(a)$ is the Gamma function and $\Gamma(a,x)$ is the incomplete Gamma function.

This question is motivated by the following [inequality][1] 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.

[1]: https://math.stackexchange.com/q/3522281/729207