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Math Help - An inequality with a product of two sums

  1. #1
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    An inequality with a product of two sums

    I'm looking a problem that states:

     \sum ^n _{i=1} x_i \sum ^n_{j=1} \frac {1}{x_j} \geq n^2

    Now if I write them down term by term I would have the following:

     x_1 ( \frac {1}{x_1} + \frac {1}{x_2} + ... + \frac {1}{x_n} ) + . . . +  x_n ( \frac {1}{x_1} + \frac {1}{x_2} + ... + \frac {1}{x_n} )

    which equals to  \frac {x_1}{x_1} + \frac {x_1}{x_2} + . . . + \frac {x_1}{x_n} + . . . + \frac {x_n}{x_n}

    which is equals to  \frac {x_1}{x_1} + . . . + \frac {x_n}{x_n} + \frac {x_1}{x_2} + \frac {x_1}{x_3} + . . . + \frac {x_n}{x_{n-1}}

    = n + \frac {x_1}{x_2} + \frac {x_1}{x_3} + . . . + \frac {x_n}{x_{n-1}}

    But how would I show that the above expression is bigger than n^2?

    I tried to use the Holder's inequality by applying the square roof on both side, but then I have:

     ( \sum ^n _{i=1} x_i)^{ \frac {1}{2} } )( \sum ^n_{j=1} \frac {1}{x_j} )^{ \frac {1}{2} } \geq n

    But I'm still stuck as the p and q are the same in this case.

    Any hints? Thank you very much!
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  2. #2
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    You already have the answer, what's the problem with Hölder's inequality?

    To put it in easier terms apply the Cauchy-Schwartz inequality to the vectors a=(\sqrt{x_1},...,\sqrt{x_n}),\ b=(\frac{1}{\sqrt{x_1}},...,\frac{1}{\sqrt{x_n}}) where you give \mathbb{R}^n the usual interior product and norm.
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