Let $0<a \le 1, \alpha<0$ 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 $\beta <1$ and increasing for $\beta>1$. $\Gamma(a)$ is the Gamma function and $\Gamma(a,x)$ is the incomplete Gamma function.

This question is a kind of generalization of the following [inequality][1], and it is migrated from mathstack after many days without answer.

I tried the approach in the above link but doesn't work in this case. By drawing the graph for some values with mathematica we can expect that the result is true. Also the sign of derivative is more delicate.

In this [question][2] I was confused with the right parameters because of the first link which affected the answer.

Maybe one would have a smart idea to do it.

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

[2]: https://mathoverflow.net/q/352103/149793