Discussion
Which of the following is not possible for an analytic function \(f\) on \(D = \{z : |z| \lt 1\}\) ?
Answer: A
The impossible case is \(f(D) = \{z : |z| \le 1\}\).
By the open mapping theorem, a non-constant analytic function on a domain maps it onto an open set, and a constant function maps it to a single point. The closed disc is neither open nor a single point, so it cannot be \(f(D)\).
The others are possible: \(f(z) = z\) gives \(f(D) = D\); \(f(z) = z/2\) gives the disc of radius 1/2; and any real constant function gives \(f(D) \subseteq \mathbf{R}\).
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