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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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    and if $z\neq -a$?2012-10-01
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    @Andres Still not I'm afraid - example: $a=i,\;z=-i/2$2012-10-01
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    Julien and this inequality is true?2012-10-01
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    Julien and this inequality is true?\\\\ $$|a_n|\leq\left|a_n+\frac{a_{n-1}}{z}+\frac{a_{n-2}}{z^2}+...+\frac{a_{1}}{z^{n-1}}+\frac{a_{0}}{z^{n}}\right|$$ with $z\neq 0$, $a_n\neq 0$ and $a_k\in \mathbb{C}$ for $k=0, 1,...,n$2012-10-02
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    Now if Julien!! I'm trying to make a proof of the fundamental theorem of algebra using the polynomial $ P (z) = a_ {0} + a_ {1} z + ... + a_ {n} z ^ {n} $ with $ a_ {n } \neq 0 $. and let me know if the above inequality is true2012-10-02
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    @Andres Still false. Let $a_n=1,\; a_{n-1}=-1,\;a_{n-2},\;a_{n-3}\cdots\; a_0=0,$ and $z=1$2012-10-02
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    and if $|z|\rightarrow\infty$ the inequality is true?2012-10-02
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    @Andres Well then you would be left with $|a_n|=|a_n|$ - so yes it would be true. I don't see how that limit relates to the fundamental theorem of algebra though.2012-10-02
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    Or did you mean for large $|z|$? (in which case it is still false)2012-10-02
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    Julien you know under what conditions the inequality $$|a_n|\leq\left|a_n+\frac{a_{n-1}}{z}+\frac{a_{n-2}}{z^2}+...+\frac{a_{1}}{z^{n‌​-1}}+\frac{a_{0}}{z^{n}}\right|$$ 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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    and if $|z|\rightarrow\infty$?2012-10-02
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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