Complex variable inequality

davesface · #1 $Old$ February 7th, 2010, 09:01 PM

Verify that $\sqrt[ ]{2} \left |z \right | \geq \left | Re(z) \right | + \left | Im(z) \right |$

Suggestion: reduce this inequality to ${( \left | x \right | - \left | y \right |)}^{2} \geq 0$

My steps:
$\sqrt[ ]{2} \left |z \right | \geq \left | x \right | + \left |y \right |$

$2 {\left |z \right |}^{2} \geq {(\left | x \right | + \left |y \right |)}^{2}$

$2 ({ x}^{2 } + { y}^{ 2}) \geq {\left | x \right | }^{ 2} + 2 \left | x \right | \left | y \right | + { \left | y \right |}^{2 }$

Where to go from here?

Opalg · #2 $Old$ February 8th, 2010, 08:01 AM

Quote:

Originally Posted by davesface $View Post$

Verify that $\sqrt[ ]{2} \left |z \right | \geq \left | Re(z) \right | + \left | Im(z) \right |$

Suggestion: reduce this inequality to ${( \left | x \right | - \left | y \right |)}^{2} \geq 0$

My steps:
$\sqrt[ ]{2} \left |z \right | \geq \left | x \right | + \left |y \right |$

$2 {\left |z \right |}^{2} \geq {(\left | x \right | + \left |y \right |)}^{2}$

$2 ({ x}^{2 } + { y}^{ 2}) \geq {\left | x \right | }^{ 2} + 2 \left | x \right | \left | y \right | + { \left | y \right |}^{2 }$

Where to go from here?

Remember that x and y are real numbers, so that $x^2 = |x|^2$ and $y^2 = |y|^2$ . If you then take everything over to the left side of that last inequality, it becomes $\bigl(|x|-|y|\bigr)^2\geqslant0$ , which is obviously true.

davesface · #3 $Old$ February 8th, 2010, 06:22 PM

Wow, that was blindingly obvious. Thanks for the tip.