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I need the proof of the following inequality. $$‎\prod‎_{i=1}^{n} {(b_i -a_i)^{1/n}} +1‎\geq \prod‎_{i=1}^{n} {(b_i)^{1/n}} +\prod‎_{i=1}^{n} {(1-a_i)^{1/n}},$$ when $0<a_i<b_i<1$.

mookid
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user135413
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Counterexample: $n=2$, $a_1\to 0$, $b_1\to 1$, $a_2\to\frac12^-$, $b_2\to\frac12^+$. Then the LHS tends to 1 but the RHS tends to $\sqrt2$.

(By the way, the variant with $a_i=b_i$ was question A2 on the 2003 Putnam, and appeared here as this question.)

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