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I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that the following inequalities hold:

$$Q(s,s+f_1(x)\sqrt{s}+f_2(x))=\frac{\Gamma(s,s+f_1(x)\sqrt{s}+f_2(x))}{\Gamma(s)}\geq e^{-x}\\ P(s,s-g_1(x)\sqrt{s}-g_2(x))=\frac{\gamma(s,s-g_1(x)\sqrt{s}-g_2(x))}{\Gamma(s)}\geq e^{-x}$$

where $s\gg0$, $x\geq0$, $\Gamma(s,y)=\int_y^\infty t^{s-1}e^{-t}dt$ and $\gamma(s,y)=\int_0^y t^{s-1}e^{-t}dt$ are the upper and the lower incomplete Gamma functions, $\Gamma(s)=\int_0^\infty t^{s-1}e^{-t}dt$ is "complete" Gamma function, and $Q(s,y)$ and $P(s,y)$ are therefore the upper and the lower regularized incomplete Gamma functions.

If the above lower bounds do not exist, I am interested in other lower bounds... Can anyone help?

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