# Why this function is monotonic?

Let $$a> 0, \alpha<0$$ 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 monotonic.

I tried the sign of derivative but is more delicate.

The function $$f$$ is not increasing for $$a=6/5>1$$, $$\alpha=-2<0$$, and $$\beta=3/4>0$$.