complex inequality

[smyroosh]

How to prove the following inequality: for complex z such that Re z < 0 :

\left| e^z-1\right| < \left| z\right| ?

[elibj123]

use the identities

\left|z\right|^{2}=z\bar{z}

and

\bar{e^{z}}=e^{\bar{z}}

since both sides of the inequality are positive
you can square it up

[smyroosh]

What next?