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- Thread starter Poirot
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- Feb 29, 2012

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Well, you could note that $(|z| - |w|)^2 \geq 0$ and work from there.

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- Feb 29, 2012

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There is the alternative of using that

$$|z|^2 + |w|^2 = \Re^2 (z) + \Im^2 (z) + \Re^2 (w) + \Im^2(w)$$

and

$$2|z||w| = 2 \sqrt{\Re^2 (z) + \Im^2 (z) } \sqrt{\Re^2 (w) + \Im^2 (w)} = 2 \sqrt{(\Re^2 (z) + \Im^2 (z))(\Re^2 (w) + \Im^2 (w))}.$$

Now, $a = \Re^2 (z) + \Im^2 (z)$ is a positive real number, same as $b = \Re^2 (w) + \Im^2 (w)$. This is the arithmetic geometric mean written in a different way:

$$\frac{a+b}{2} \geq \sqrt{ab} \implies \frac{\Re^2 (z) + \Im^2 (z) + \Re^2 (w) + \Im^2(w)}{2} \geq \sqrt{(\Re^2 (z) + \Im^2 (z))(\Re^2 (w) + \Im^2 (w))}.$$

How did you get that $(|z| - |w|)^2 = |z|^2 + |w|^2 - 2 \Re (z \bar{w})$?