Abs(e^z-1) less than or equal to e^(abs(z))-1 less than or equal to abs(z)e^(abs(z))

Theorem

The following formula holds for all $z \in \mathbb{C}$: $$\left| e^z - 1 \right| \leq e^{|z|}-1 \leq |z| e^{|z|},$$ where $e^z$ denotes the exponential.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.2.39