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Ramanujan presented this double inequality without showing a proof, namely: $\sqrt\pi \left(\frac{x}{e}\right)^{x}\left(8x^3+4x^2+x+\frac{1}{100} \right)^{1/6}<\Gamma(x+1)<\sqrt\pi \left(\frac{x}{e}\right)^{x}\left(8x^3+4x^2+x+\frac{1}{30} \right)^{1/6}, \space x \ge 0$ I wonder if there is a nice proof for it. Could you help here? Thanks.

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    @daniel: I know the problem from my brother Chris and he knows it from his teacher. I can't tell you more.2012-09-03

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It is probably explained in this paper or its references.

http://arxiv.org/pdf/1112.1896

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    Nice paper! (+1)2012-09-03