If $a_1a_2\cdots a_n =1$, then $\prod_{i=1}^n (1+a_i) \geq 2^n$

Question

Let $$ a_1,a_2,\dots,a_n $$ be positive real numbers such that $$ \prod_{i=1}^n a_i =1 $$ Prove that $$ \prod_{i=1}^n (1+a_i) \geq 2^n $$

Answer 1

Hint: note that $\frac{1+a_i}{2} \geq \sqrt{a_i}$ (by the AM-GM inequality), so that $$ \prod_{i=1}^n \left(\frac{1+a_i}{2}\right) \geq \prod_{i=1}^n \sqrt{a_i} $$

Answer 2

By Holder $$\prod_{i=1}^n(1+a_i)\geq\left(1+\sqrt[n]{\prod\limits_{i=1}^na_i}\right)^n=2^n$$