Bounds for Finite Product of Real Numbers

Theorem
Let $a_1, a_2, \ldots, a_n$ be positive real numbers.

Then:
 * $\displaystyle \sum_{k \mathop = 1}^n a_k \le \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \map \exp {\sum_{k \mathop = 1}^n a_k}$

Lower bound
Follows by expanding.

Proof 1
By Exponential of x not less than 1+x:


 * $\displaystyle \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \prod_{k \mathop = 1}^n \exp a_k = \map \exp {\sum_{k \mathop = 1}^n a_k}$

Proof 2
By the AM-GM Inequality:


 * $\displaystyle \prod_{k \mathop = 1}^n \paren {1 + a_k} \le \paren {\frac {n + \sum_{k \mathop = 1}^n a_k} n}^n$

Also see

 * Weierstrass Product Inequality