# Two Inequalities Involving Complex Numbers

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I have two problems with complex numbers:

1) Use properties of moduli to sow that when |z3| does not equal |z4|,

Re(z1 + z2) / |z3 + z4| <= (is smaller or equals) (|z1| + |z2|) / (| |z3| -|z4| |)

2) Verify that sqrt(2) * |z| >= |Re z| + |Im z|

(suggestion: reduce this inequality to (|x| - |y|)^2 +. 0

https://brainmass.com/math/complex-analysis/two-inequalities-involving-complex-numbers-456774

#### Solution Preview

1) We have

Re(z1 + z2) <= |z1 + z2| <= |z1| + |z2|

and

|z3 + z4| >= ||z3| - |z4||,

whence ...

#### Solution Summary

We prove two inequalities involving complex numbers.

$2.49