Assume f(z) is analytic in a domain R and never vanishes, but for some $z_{o}$ in R: $|f(z_{o})|=min(|f(z)|)$ in R. How to prove that f is constant?

Question

I tried solving this question by taking the derivative of f(z) = u(x,y) +iv(x,y). Then setting the derivative to zero at $z_o$. But doing this did not give any useful insight. I am generally confused by how I can relate f(z) never vanishing and being constant but there being a minimum value among the values of |f(z)|.