# inequality proof

• Sep 10th 2008, 11:06 PM
lllll
inequality proof
I'm a little stuck with the following inequality and was wondering if anyone could help me out.

$\displaystyle \frac{\mbox{Re}(z_1+z_2)}{|z_3+z_4|} \leq \frac{|z_1|+|z_2|}{||z_3|-|z_4||} \ \ \mbox{where} \ \ z_3 \neq z_4$

$\displaystyle \frac{x_1+x_2}{|z_3+z_4|} \leq \frac{|z_1|+|z_2|}{||z_3|-|z_4||}$

$\displaystyle \frac{x_1+x_2}{|z_3+z_4|} \leq \frac{\sqrt{x_1^2+y_1^2}+\sqrt{x_2^2+y_2^2}}{||z_3 |-|z_4||}$

at which point I get stuck.

I was also thinking of using the property that:

$\displaystyle |z_3+z_4| \geq ||z_3|-|z_4|| \therefore \frac{1}{|z_3+z_4|} \leq \frac{1}{||z_3|-|z_4||}$ but the numerators are throwing me off.
• Sep 10th 2008, 11:49 PM
Moo
Hello,

Quote:

I was also thinking of using the property that:

$\displaystyle |z_3+z_4| \geq ||z_3|-|z_4||$
What is this property ? :confused:

Quote:

but the numerators are throwing me off.
By the numerators, were you thinking of $\displaystyle x_1+x_2 \le \sqrt{x_1^2+y_1^2}+\sqrt{x_2^2+y_2^2}$ ?
$\displaystyle \sqrt{x_1^2+y_1^2}+\sqrt{x_2^2+y_2^2} \ge \sqrt{x_1^2}+\sqrt{x_2^2}=|x_1|+|x_2| \ge x_1+x_2$
• Sep 11th 2008, 02:26 AM
Laurent
For the numerator, use Moo's idea, and for the denominator, use indeed yours (the lower bound in the triangular inequality).

About this lower bound, it is a consequence of the usual triangle inequality: for any $\displaystyle x,y$, $\displaystyle |x|\leq|x+y|+|-y|$, hence $\displaystyle |x|-|y|\leq |x+y|$ and by symmetry the same holds swapping $\displaystyle x$ and $\displaystyle y$, so that $\displaystyle ||x|-|y||=\max(|x|-|y|,|y|-|x|)\leq |x+y|$.

Laurent.