Helmholtz equation Inequality (potential theory)

Question

I was reading the chapter about surface potential. I came across an inequality

$$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|,$$

$k$ is a complex number here. Since $k$ is a complex number here, I am a bit confused. I am wondering what is the correct way to interpret this ineuqality?

Answer 1

The map $z \mapsto e^z$ maps complex numbers to complex numbers. Hence $e^{ik|x_1-y|}$ is a complex number and

$$|e^{ik|x_1-y|}-e^{ik|x_2-y|}|\leq k|x_1-x_2|$$ is an inequality between complex numbers modulus.