I need to prove that $\int_\gamma f'(z)/f(z)dz=0$ for any closed curve. It is given that f is holomorphic and satisfies $|f(z)-1|\lt1$ in the region. And we can assume $f'(z)$ is continuous.
I think that first I need to prove that $f'(z)/f(z)$ has a primitive, so that I can use the proposition to prove the integral is equal to 0.
But I don't know how to prove $f'/f$ has a primitive and how to use that inequality in my proof? Thanks.