modulus of a complex number

Aug 2011
143
1
In one of the complex number problems which I was working with

It was given as follows:

|z^3 + z^-3| <= |z|^3 + 1/[|z|^3]

How is it?

I am not able to figure it out.

guidance is required.

with regards

aranga
 

Plato

MHF Helper
Aug 2006
22,490
8,653
It was given as follows:
\(\displaystyle |z^3 + z^{-3}| <= |z|^3 + 1/[|z|^3]\)
How is it?
If you are studying complex numbers, then you know that for all numbers \(\displaystyle z~\&~w\):
the triangle inequality holds \(\displaystyle |z+w|\le|z|+|w|\),
\(\displaystyle |z^3|=|z|^3\), and \(\displaystyle \left|\dfrac{1}{w}\right|=\dfrac{1}{|w|}(w\ne 0)\).
By using those simple properties we can get your result.
 
Aug 2011
143
1
Thank you very much. It is very useful.

If you are studying complex numbers, then you know that for all numbers \(\displaystyle z~\&~w\):
the triangle inequality holds \(\displaystyle |z+w|\le|z|+|w|\),
\(\displaystyle |z^3|=|z|^3\), and \(\displaystyle \left|\dfrac{1}{w}\right|=\dfrac{1}{|w|}(w\ne 0)\).
By using those simple properties we can get your result.