$|e^{z_1}-e^{z_2}|$ is less than $|z_1-z_2|$ if the real parts are non-positive

Question

Is $|e^{z_1}-e^{z_2}| \le|z_1-z_2|$ if the real parts of $z_1,z_2$ are non-positive?

I think yes, but what method do I proceed with? Do I proceed with the fact that modulus equals the number multiplied by its conjugate, or do I expand the complex numbers into real and imaginary parts. Any hints? Thanks beforehand.

Answer 1

Hint: $f(z_1) - f(z_2) = \int_C f'(z)\; dz $ where $C$ is the directed line segment from $z_2$ to $z_1$, so $|f(z_1) - f(z_2)| \le |z_1 - z_2| \sup_{z \in C} |f'(z)|$.