How to prove the following statement?
For any $x > 0$, $\gamma(x+1,x) < \Gamma(x+1,x+1) < \gamma(x+1,x+1) < \Gamma(x+1,x)$
or, equivalently,
$$\int_0^x t^x e^{-t} dt < \int_{x+1}^\infty t^x e^{-t} dt < \int_0^{x+1} t^x e^{-t} dt < \int_x^\infty t^x e^{-t} dt$$
where $\gamma,\Gamma$ are the lower and upper incomplete $\Gamma$ function, respectively.
The first inequality translates to $\Gamma(x+1) - \Gamma(x+1,x) < \Gamma(x+1,x+1)$, so $$ \int_0^\infty t^x e^{-t} dt < 2\int_{x+1}^\infty t^x e^{-t} dt + \int_x^{x+1} t^x e^{-t} dt$$ $$ \int_0^{x+1} t^x e^{-t} dt < \int_x^\infty t^x e^{-t} dt$$ which turns out to be the 3rd inequality. Change of variable does not seem to help here. What should I do to solve this?