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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.

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  • $\begingroup$ Do you want your inequality to hold for any $x>0$ or just for any $x$ large enough? $\endgroup$ – Jack D'Aurizio Sep 18 '17 at 18:53
  • $\begingroup$ Anyway, my previous remark still stands. Since $t^{x\log 2}e^{-t}$ is concentrated around $t=x\log 2 \ll x$, I would expect the LHS to converge to $1$ as $x\to +\infty$, violating your inequality for any $b>0$. $\endgroup$ – Jack D'Aurizio Sep 18 '17 at 18:55
  • $\begingroup$ I think that if one shows it for large enough $x$ than it should be possible to adjust the coefficients to account for smaller $x$ $\endgroup$ – Boris K. Sep 18 '17 at 18:57
  • $\begingroup$ Thank you, instead of $\ln 2$ it should have been $\log_2 e=(\ln 2)^{-1}$. Apparently that should hold or for any constant greater than 1. $\endgroup$ – Boris K. Sep 18 '17 at 19:19
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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

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  • $\begingroup$ Thank you, that looks like what I was looking for! $\endgroup$ – Boris K. Sep 18 '17 at 18:51

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