Inequality of complex numbers modulus

Question

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

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

Please do something about your accept rate: meta.stackoverflow.com/questions/5234/…

Ross Millikan · Answer 1 · 2012-10-01 22:50:27Z

up vote 3 down vote

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

answered Oct 1 '12 at 22:50

Ross Millikan
83.2k555117

score 1 · Answer 2 · 2012-10-01 23:15:09Z

up vote 1 down vote

It isn't true.

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

edited Oct 1 '12 at 23:15

answered Oct 1 '12 at 22:51

user39572

	and if $z\neq -a$? – Roiner Segura Cubero Oct 1 '12 at 23:40
	@Andres Still not I'm afraid - example: $a=i,\;z=-i/2$ – user39572 Oct 1 '12 at 23:47
	Julien and this inequality is true? – Roiner Segura Cubero Oct 1 '12 at 23:56
	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 $k=0, 1,...,n$ – Roiner Segura Cubero Oct 2 '12 at 0:04
	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 true – Roiner Segura Cubero Oct 2 '12 at 0:11

mike4ty4 · Answer 3 · 2012-10-02 01:40:31Z

up vote 0 down vote

This is false(!) Try $a = 1$, $z$ any negative number in $[-2, 0]$.

edited Oct 2 '12 at 1:40

answered Oct 2 '12 at 1:12

mike4ty4
1,43029

	and if $\|z\|\rightarrow\infty$? – Roiner Segura Cubero Oct 2 '12 at 1:30
	For larger $z$, then your inequality can hold. But not for all $z$. But you said "$\forall z \in \mathbb{C}$". – mike4ty4 Oct 2 '12 at 1:43

3 Answers