I would like to bound from above the expression

$$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)} $$ for $x>y>0$. By plotting the above expression I have found that it should hold $$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}< y^{\alpha-\beta} $$ for $\alpha > \beta$ and $$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}< x^{\alpha-\beta} $$ for $\alpha < \beta$. To prove the first estimate my guess was to show that a function $$ f(\alpha)=y^{-\alpha}(\Gamma(\alpha,x)-\Gamma(\alpha,y)) $$ is increasing in $\alpha$. It seemed natural to me to compute $\partial_\alpha f(\alpha)$ but things started to get rough because $\partial_\alpha \Gamma(\alpha,x)$ has to be expressed as Meijer G-function. So the question is is there a more elementary proof of the presented estimates? Are there "better" estimates of which my estimates are special cases?

I would like to bound from above the expression

$$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)} $$ for $x>y>0$. By plotting the above expression I have found that it should hold $$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}< y^{\alpha-\beta} $$ for $\alpha < \beta$ and $$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}< x^{\alpha-\beta} $$ for $\alpha > \beta$. To prove the first estimate my guess was to show that a function $$ f(\alpha)=y^{-\alpha}(\Gamma(\alpha,x)-\Gamma(\alpha,y)) $$ is increasing in $\alpha$. It seemed natural to me to compute $\partial_\alpha f(\alpha)$ but things started to get rough because $\partial_\alpha \Gamma(\alpha,x)$ has to be expressed as Meijer G-function. So the question is is there a more elementary proof of the presented estimates? Are there "better" estimates of which my estimates are special cases?