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Let $a> 1, \alpha<0$ 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 $a<1$ and increasing for $a>1$.

This question is motivated by the following inequality after drawing the graph for some values with wolfram. I tried the approach in the above link but doesn't work in this case. Also the sign of derivative is more delicate.

Maybe one would have a simple idea, but any suggestion would be helpful.

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.

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

This question is motivated by the following inequality after drawing the graph for some values with wolfram. I tried the approach in the above link but doesn't work in this case. Also the sign of derivative is more delicate.

Maybe one would have a simple idea, but any suggestion would be helpful.

Let $a> 1, \alpha<0$ 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 $a<1$ and increasing for $a>1$.

This question is motivated by the following inequality after drawing the graph for some values with wolfram. I tried the approach in the above link but doesn't work in this case. Also the sign of derivative is more delicate.

Maybe one would have a simple idea, but any suggestion would be helpful.

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 motivated by the following inequality after drawing the graph for some values with wolfram. I tried the approach in the above link but doesn't work in this case. Also the sign of derivative is more delicate.

Maybe one would have a simple idea, but any suggestion would be helpful.

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

This question is motivated by the following inequality after drawing the graph for some values with wolfram. I tried the approach in the above link but doesn't work in this case. Also the sign of derivative is more delicate.

Maybe one would have a simple idea, but any suggestion would be helpful.