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Math Help - Prove an inequality

  1. #1
    puch7524
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    Prove an inequality

    Can anyone prove the following inequality:

    <br />
\sum_{i=1}^N y_{i}^2/n_{i} - \frac{\left(\sum_{i=1}^N y_{i}\right)^2}{n} \ge 0,<br />

    where n=\sum_{i=1}^N n_{i},\ \ n_{i} \ge 0, and y_{i} is a real quantity (can be both negative or positive)?

    It appears that the inequality is valid (even used random numbers), but can't see how to prove it.
    Last edited by CaptainBlack; June 6th 2007 at 10:41 AM.
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  2. #2
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    Quote Originally Posted by puch7524 View Post
    Can anyone prove the following inequality:

    <br />
\sum_{i=1}^N y_{i}^2/n_{i} - \frac{\left(\sum_{i=1}^N y_{i}\right)^2}{n} \ge 0,<br />

    where n=\sum_{i=1}^N n_{i},\ \ n_{i} \ge 0, and y_{i} is a real quantity (can be both negative or positive)?

    It appears that the inequality is valid (even used random numbers), but can't see how to prove it.
    This inequality gets really messy really quickly. So I will prove the special case N=2. I am sure you can use the same method to generalize it to more terms but it just is so messy.
    ---
    First, I think you meant n_i >0.

    Thus, we have to show:
    \frac{y_1^2}{n_1}+\frac{y_2^2}{n_2} \geq \frac{(y_1+y_2)^2}{n_1+n_2}

    Rewrite as,
    \frac{y_1^2}{n_1}+\frac{y_2^2}{n_2} \geq \frac{y_1^2}{n_1+n_2}+\frac{y_2^2}{n_1+n_2}+ 2\cdot \frac{y_1y_2}{n_1+n_2}

    Multiply by n_1n_2(n_1+n_2)>0:

    n_2(n_1+n_2)y_1^2+n_1(n_1+n_2)y_2^2 \geq y_1^2n_1n_2+y_2^2n_1n_2+2y_1y_2n_1n_2

    Open and cancel,

    n_2^2y_2^2 + n_1^2y_2^2 \geq 2y_1y_2n_1n_2

    This is the AM-GM inequality.
    Which is true.

    (Or you can write (n_1y_1 - n_2y_2)^2 \geq 0).
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