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As I read a paper, I don't know how do I derive inequality, $$\frac{\prod^N_{i=1}4^{b_i/N}(1-4^{-b_i})^{1/N}}{12}\ge \frac{4^{R/N}}{16} \\ \frac{\prod^N_{i=1}(4^{b_i}-1)^{1/N}}{12}\ge \frac{4^{R/N}}{16}.$$ where $\sum^N_{i=1}b_i=R$.

I think it just need a simple idea but I've not found way so please help me to do something.

Thank you.

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Is $R \le N$? Are $b_i$ positive? You may need to give more background to this. –  Macavity Mar 11 at 18:44

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