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Math Help - Complex Inequalities

  1. #1
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    Complex Inequalities

    Suppose that z and w are in the closed unit disk.

    Prove the inequality 1- \vert w \vert^2 \leq 2 \vert 1-z \bar w \vert.

    I've tried to square both sides and expand the equation. However, I'm stuck there. Is there any other way that I can approach this question?

    Thanks in advance.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint :

    Use

    |1-z\bar{w}|\geq |\;1-|z\bar{w}|\;|=1-|z||\bar{w}|\geq \ldots


    Fernando Revilla
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  3. #3
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    2 \vert 1- z \bar w \vert<br />
\geq 2(\vert 1- \vert z \bar w \vert \vert )<br />
\geq 2(1- \vert zw \vert )<br />
\geq 2(1- \vert w \vert )(Since \vert z \vert \leq 1)
    Now we claim that 2 \vert w \vert - \vert w \vert^{2} \leq 1.
    We let y=2 \vert w \vert - \vert w \vert^{2} and substitute \vert w \vert with x, where 0 < x \leq 1. Then we differentiate the equation y to get the maximum point. From here we can get y \leq 1. Hence 2 \vert w \vert - \vert w \vert^{2} \leq 1.
    Therefore, 2 \vert 1- z \bar w \vert<br />
\geq 2(1- \vert w \vert )<br />
\geq 2(1- \frac{1+ \vert w \vert^2}{2})<br />
=1- \vert w \vert^2.

    Is there any easier method to this problem?
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