Extension of a real function to the complex plane should carry regularity properties

I have a function g:R^+ -> R, that is Lipschitz continuous and g(0)=0. I extend it over the complex plane, by

g(z)= (z/|z|)*g(|z|).

Now I try to prove directly that the extension is Lipschitz continuous, but I can't get past the first steps!

If someone could help me, please. Just try to prove it and you'll see what I mean.

Thx