Given that a function f is holomorphic in the open unit disk D = {z ∈ C: |z| < 1} and satisfies the conditions f(0) = 0 and |f(z)| ≤ 1 for all z in D. Consider another function g defined as g(z) = f(z) - φz where φ is a complex number such that |φ| = 1. If for some z ≠ 0 in D, |f(z)| = |z|, which of the following statements is true for g:
1
g(z) = 0 for all z in D
2
g(z) is a non-zero constant for all z in D .
3
g(z) is at least an isometry with respect to the Poincaré metric
4
|g'(0)| ≤ 1