I am trying to prove that $$\frac{\gamma(x/ \ln 2 ,x)}{\Gamma(x / \ln 2)}<ae^{-bx}$$ for some positive $a$ and $b$, where $\gamma(x,s) = \int_0^s t^{x-1}e^{-t}dt$ is lower incomplete gamma function.
Asymptotics of $\Gamma(x)$ is well known but I struggle to come with good bounds for the incomplete gamma.
The inequality from http://dlmf.nist.gov/8.10.E11 might help $$(1-e^{-\alpha_{a}x})^{a}\leq P\left(a,x\right)\leq(1-e^{-\beta_{a}x})^{a},$$ with $$P(a,x) = \frac{\gamma(a,x)}{\Gamma(a)}$$ and $\alpha_{a}, \beta_{a}$ given by http://dlmf.nist.gov/8.10.E12