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

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.


Maybe one would have a smart idea to do it.

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