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.