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.


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 $f= \infty$.

  • $\begingroup$ 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. $\endgroup$ – Chris Mar 6 '13 at 7:49
  • $\begingroup$ @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. $\endgroup$ – Hans Mar 6 '13 at 10:08
  • $\begingroup$ @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? $\endgroup$ – Chris Mar 6 '13 at 10:33
  • $\begingroup$ @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. $\endgroup$ – Hans Mar 6 '13 at 12:34
  • $\begingroup$ @Chris: see my edit. $\endgroup$ – Hans Mar 6 '13 at 13:06

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