Arranging set A and B to maximize their power

Question

Given two sets A and B each with $n$ positive reals.

How to arrange elements in A and B such that

$$\prod_{i=1}^n a_i^{b_i}$$

is maximized?

Will ascending order of A and B make the correct answer? Please explain.

Answer 1

Suppose that $a_1\lt a_2$ and $b_1\lt b_2$ then $$ \frac{a_1^{b_1}a_2^{b_2}}{a_1^{b_2}a_2^{b_1}}=\left(\frac{a_2}{a_1}\right)^{b_2-b_1}\gt1 $$ Thus, simply matching the order of two elements that are out of order with each other makes the product bigger. This implies that if both are sorted in the same order, the product is maximized (otherwise, switch any two that are out of order with each other, and the product increases). Therefore, you are correct.

Answer 2

Let $P = \prod_{i=1}^n a_i^{b_i} $ and $p = \sum_{i=1}^n b_i \ln a_i $ where the $a_i$ is a non-decreasing sequence. We want the arrangement of the $b_i$ which maximizes $P$.

Maximizing $P$ is the same as maximizing $p$.

Then Chebychev's inequality (this one: http://en.wikipedia.org/wiki/Rearrangement_inequality) says that $p$ is max when the $b_i$ are non-decreasing and min when the $b_i$ are non-increasing.

So your conjecture is correct.