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Thread: 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 $\displaystyle X,Y,Z,W>0$

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

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

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

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

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

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

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