MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top


What is the maximum of the following function?:

$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ x_i}\right )} $


$i = 1,..,n$

$1 < w_i$

$0 < x_i < 1$

The generalization of the inequality of arithmetic and geometric means may indicate that the maximum is reached, when all $x_i$ are equal (the minuend is a weighted harmonic mean) but I haven't managed to proove it.

EDIT: The above question was badly constructed. The answer I am looking for is: For what values of $w_i$ and $x_i$, in case of any $i=1,..,n$ the following function reaches maximum.

EDIT2: I am looking for the optimum distribution of $w_i$ and $x_i$. The specific values of $w_i$ and $x_i$ are not needed. The problem needs to be solved, knowing that the minuend is a set number: $S=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} }$ . So actually the question can be redefined: for given S, what is the optimum distribution of $w_i$ and $x_i$ so $f(x_i,w_i)$ reaches maximum.

share|cite|improve this question

For $n=1$ you get $f=0$ for all values.

For $n=2$ you get for $w_1 = w_2 \approx 1$ and $x_1 = x_2 = 1/2$ division by zero in the second term, and so $f= \infty$.


For every even $n$ you get division by zero for $w_1 = \ldots = w_n \approx 1$ and $x_1 = \ldots = x_n = 1/2$, and so $f= \infty$.

share|cite|improve this answer
Hi Hans, thank you for your answer. However, i asked a wrong question (will edit it now). I am not searching the maximum value - I am searching in what conditions (for what values of $x_i$ and $w_i$, for any $i=1,..,n$) the maximum is reached. That is why, I mentioned the inequality of arithmetic and geometric means, as it may indicate that the maximum is reached when all $x_i$ are equal, but I don;t know how to proove it. – Chris Mar 6 '13 at 7:49
@Chris: but your problem seems ill-posed: already for $n=2$ you get $f=\infty$ for the values I state above. This can be already guessed as your domain is not compact, and so f need not attain a maximum. - Please let the original post, and write your reformulation as "Ed: .." as an addition to your present post. – Hans Mar 6 '13 at 10:08
@Hans: I returned to the original version of the question adding the EDIT part, as you requested. I agree, for $n=2$, when $w_1 = w_2$ and $x_1 = x_2$ the function f reaches infinity, but can it be proven that that for any $i=1,..,n$ keeping $w_1 = ... = w_n$ and $x_1 = ... = x_n$ the function f will always reach maximum? – Chris Mar 6 '13 at 10:33
@Chris: your edit has not changed the message of your post at all. This I have answered. Your last comment says you have only two parameters, namely $w$ (= $w_i$ for all $i$) and $w/x$ (= $w_i/x_i$ for all $i$), which you could easily discuss by yourself. – Hans Mar 6 '13 at 12:34
@Chris: see my edit. – Hans Mar 6 '13 at 13:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.