- #1

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## Homework Statement

Let f and g be two holomorphic functions in the unit disc D1 = {z : |z| < 1}, continuous in D1, which

**do not vanish**for any value of z in the closure of D1. Assume that |f(z)| = |g(z)| for every z in the boundary of D1 and moreover f(1) = g(1). Prove that f and g are the same function.

## Homework Equations

Maximum modulus?

## The Attempt at a Solution

Last one for now.

Okay I had enough sense to gather from the hint that f and g don't vanish that I should define h = f/g. Then h(1) = 1, |h(z)| = 1 for z on the boundary of D1. Now I feel like maximum modulus will give that h(z) = 1 on D1, but I'm making a logical leap. Can someone help me out here? Thanks.