If f and g are continuous functions of a real variable and are non-zero in some interval, then clearly |f(x)| = |g(x)| implies f(x) = g(x) or f(x) = -g(x) on that interval.
What is the analog for functions of a complex variable?
This is my suspicion: if f and g are nonvanishing analytic complex-valued functions on some connected subset D of C, and |f(z)| = |g(z)| for all z in D, what can be said of f and g? Is it necessary that f = cg, where |c| = 1? If this is right, how would one prove it?
Thanks in advance for any help.
I meant to say f and g are continuous (real variable case) and analytic (complex variable case) respectively.
I don't think that counterexample you give is analytic, so I'm still thinking it could be true.