If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Question

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers.

Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$.

I don't understand why that is. Any help would be greatly appreciated.

Answer 1

Hint: $\bigl||x|-|y|\bigr|\leq |x-y|$ (the reverse triangle inequality) holds for complex numbers $x,y$.