1. ## prove problem

z1/z2+z3 <= |z1|/||z2|-|z3|| ; |z2|<>|z3|, z=x+iy

thanks.

2. Originally Posted by unicorn

z1/z2+z3 <= |z1|/||z2|-|z3|| ; |z2|<>|z3|, z=x+iy

thanks.
There is an impossibility in your problem.
The right hand side is a real number because of absolute value ffor complex numbers, but the left hand remains a complex number. How can a complex number be ordered?

3. excuse me! the problem is:
|z1/z2+z3| <= |z1|/||z2|-|z3|| ; |z2|<>|z3|, z=x+iy

4. Originally Posted by unicorn
excuse me! the problem is:
|z1/z2+z3| <= |z1|/||z2|-|z3|| ; |z2|<>|z3|, z=x+iy
If you mean:

Prove that $\forall z_1, z_2, z_3 \in \mathbb{C}$:

$\frac{|z_1|}{|z_2+z_3|} \le \frac{|z_1|}{|\ |z_2|-|z_3|\ |}$?

Then it is sufficient to prove that:

$|z_2+z_3| \ge |\ |z_2|-|z_3|\ |$

RonL

5. maybe it's silly, but how can I prove that
$
|z_2+z_3| \ge |\ |z_2|-|z_3|\ |
$