The absolute value of the exponential of $z$ is bigger than the exponential of $-|z|$

Question

Let $z$ be a complex number. Is

\begin{equation} |\exp(z)| \geq \exp(-|z|) \end{equation}

for arbitrary $z$ or do we need a condition on $z$?

How can one prove this? I tried the inverse triangle inequality but couldn't do it.

Answer 1

Hint: $|\exp(z)| = \exp (\Re (z)).$ $~~~$Thus,

$\exp (\Re (z)) \geq \exp \left(-\sqrt{\Re ^2(z)+\Im^2 (z)}\right) \implies \Re (z) \geq -\sqrt{\Re ^2(z)+\Im^2 (z)}.$

$\Re (z)$ and $\Im (z)$ are the real and imaginary parts of $z$.