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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.

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The claim is incorrect. See e.g. this image of a Mathematica notebook:

enter image description here

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

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    $\begingroup$ @Migalobe : Your edits are multiple and substantial, and they invalidate the answer. In such a case, it is better to restore the original question and post the modified question separately. I'd also suggest to carefully check and recheck the question before posting it, to prevent people from wasting their time answering "wrong" questions. $\endgroup$
    – Iosif Pinelis
    Commented Feb 7, 2020 at 14:00

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