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Someone could help to prove the following inequality of modulus of complex numbers:

If $a\in\mathbb{C}$ then $|a|\leq|a+z| \qquad \forall z\in\mathbb{C}$

3 Answers 3

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It will be hard to prove. Try $a=1, z=-1$ where it is false.

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It isn't true.

$a=1+i,\;z=-1-i$

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    Julien you know under what conditions the inequality $|a_n|\leq\left|a_n+\frac{a_{n-1}}{z}+\frac{a_{n-2}}{z^2}+...+\frac{a_{1}}{z^{nā€Œā€‹-1}}+\frac{a_{0}}{z^{n}}\right|$ is true? – 2012-10-02
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This is false(!) Try $a = 1$, $z$ any negative number in $[-2, 0]$.

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    For larger $z$, then your inequality can hold. But not for *all* $z$. But you said "$\forall z \in \mathbb{C}$". – 2012-10-02