Home » Blog » Conjecture an inequality for products

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

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):