Conjecture an inequality for products

If $a_k < b_k$ for $k = 1, \ldots, n$ , what conditions are needed for the inequality

$\prod_{k=1}^n a_k < \prod_{k=1}^n b_k$

to hold?

Claim: In addition to $a_k < b_k$ we also need $0 \leq a_k$ for each $k = 1, \ldots, n$ .
Proof. The statement is certainly true for the case $n=1$ (since $a_1 < b_1$ by assumption). Assume then that it is true for some $n = m \in \mathbb{Z}_{>0}$ . Then,

$\prod_{k=1}^{m+1} a_k = a_{m+1} \cdot \prod_{k=1}^m a_k < a_{m+1} cdot \prod_{k=1}^m b_k$

by the inductive hypothesis. But then, since $b_{m+1} > a_{m+1} \geq 0$ ,

$a_{m+1} \cdot \prod_{k=1}^m b_k < b_{m+1} \cdot \prod_{k=1}^m b_k = \prod_{k=1}^{m+1} b_k.$

Thus, the statement is true for $m+1$ ; and hence, for all $n \in \mathbb{Z}_{> 0}. \qquad \blacksquare$

Stumbling Robot

Conjecture an inequality for products

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