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Suppose that $n$ is a certain Natural number and $\{b_i\}_{i=0,..,n-1}$ is a certain array of positive real numbers, find array $\{a_i\}_{i=0,..,n-1}$ of positive real numbers such that $\sum_{i=0}^{n-1} a_i = 1$ and $\sum_{i=0}^{n-1} \sqrt {a_i b_i}$ is the maxmum possible value.
I think thats maximum when all $\frac{a_i}{b_i}$ ratios are equal. But I need to make sure.

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You may not belive that this problem came out from working on a software design (a model for internet sever provider and bandwith management). I am now a programmer and I was many years far from pure mathematics, and forgot most of these iniqality techniques (I remind how much important and applicable was cauchy-schwarz) – saeedgnu Apr 10 '11 at 0:16
up vote 3 down vote accepted

You are right: by Cauchy-Schwarz inequality, $\sum\sqrt{a_ib_i}\leq\sqrt{\sum a_i}\sqrt{\sum b_i}$, with equality iff $a_i=\lambda b_i$ for some positive $\lambda$.

edit: oops, I thought you were looking for the maximum. Minimum is when all $a_i$'s are $0$, except for the $i$ where $b_i$ is minimal (so there $a_i=1$).

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Then that would be the maximum when $a_i=\lambda b_i$. Also here the sequence $(a_i)$ is not given, we need to find these $a_i's$. – user9077 Apr 9 '11 at 13:54
@user9077: I (believe I) found and described those $a_i$'s where we get a minimum (i.e. all equal to $0$ except for one) – user8268 Apr 9 '11 at 14:49
yeah. maximum. I was going to say oops!! but saw your answer. you are right and I got my answer (I edit post and change to maximum). thanks. – saeedgnu Apr 9 '11 at 16:12

Your conjecture is correct (key term: Cauchy-Schwarz Inequality).

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