The upper incomplete Gamma function is defined as $\Gamma(a,x):=\int_x^\infty t^{a-1}e^{-t}dt$ for any $a\in \mathbb{R}$ and $x\in \mathbb{R}^+$.
I am trying to find an exact, preferably tight, lower bound on $\Gamma(-1,x)$.
I have looked at this paper and it gives lower bounds for $\Gamma(a,x)$ for $a\geq 1$ in Theorem $1.1$ and for $a<-1$ in Theorem $1.2$. Any suggestions?