Proving that $ \prod_{i=1}^{n} (1 + a_{i}) \ge \left(1 + \prod_{i=1}^{n} a_{i}^{1/n} \right)^{n} $

Asked Dec 23 '14 at 22:44

Active Dec 14 '16 at 13:40

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I'd like some help on proving the following inequality $$ \prod_{i=1}^{n}\left(1+a_{i}\right) \ge \bigg(1 + \prod_{i=1}^{n} a_{i}^{1/n}\bigg)^{n}, $$ given that $ a_{i} > 0\,\, \forall\, i\in\{1,2,\cdots,n\} $.

I tried to use AM-GM inequality on the right-hand side, but without success. Any help would be appreciated.

edited Dec 14 '16 at 13:40

Martin Sleziak

56,060

asked Dec 23 '14 at 22:44

user149844

1,335

Substitute $a_k$ with $a_k^n$? – Math.StackExchange Dec 23 '14 at 22:54
Use Cauchy Shwartz inequality! – Arash Dec 23 '14 at 22:58
@AlbertSteppi, thank you very much! – user149844 Dec 23 '14 at 23:58

Proving that $ \prod_{i=1}^{n} (1 + a_{i}) \ge \left(1 + \prod_{i=1}^{n} a_{i}^{1/n} \right)^{n} $

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