Prove Complex Inequality: Re z < 0

[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?