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Math Help - Cauchy–Schwarz inequality- Question

  1. #1
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    Cauchy–Schwarz inequality- Question

    Help with that
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  2. #2
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    Quote Originally Posted by sbsite View Post
    Help with that
    Simple, since X,Y,Z,W>0

    \bold{a}=<\sqrt{X},\sqrt{Y},\sqrt{Z},\sqrt{W}>

    \bold{b}=\left< \frac{1}{\sqrt{X}},\frac{1}{\sqrt{Y}},\frac{1}{\sq  rt{Z}},\frac{1}{\sqrt{W}} \right>

    Thus the dot product is,
    \bold{a}\cdot \bold{b}=1+1+1+1=4

    Thus,
    4\leq ||\bold{a}||||\bold{b}||
    Square both sides,
    16\leq (X+Y+Z+W)\left(\frac{1}{X}+\frac{1}{Y}+\frac{1}{Z}  +\frac{1}{W} \right)
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  3. #3
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    this is also a result of

    (a_1+a_2+\dots+a_n)\left(\frac{1}{a_1}+\frac{1}{a_  2}+\dots+\frac{1}{a_n}\right)\ge n^2

    assuming that a_i>0 for all i such that 1\le{i}\le{n}

    that can be proved by either using AM-GM or Cauchy
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