Discussion
A function f analytic on a disk D is necessarily constant if f(D) is contained in :
Answer: C
An analytic \(f\) with \(f(D)\) inside a circle must be constant.
A circle is a curve with empty interior in C. By the open mapping theorem, a non-constant analytic function on a domain maps open sets to open sets, so \(f(D)\) would contain a disk, which a circle cannot. Hence \(f\) is constant. The same follows from the Cauchy-Riemann equations: if \(|f - c|\) is constant, so is \(f\).
The other sets have interior, so they impose nothing: \(f(z) = z\) maps the unit disk into a disk, \(f(z) = z + 2\) maps it into the half plane Re \(w \gt 0\), and every analytic function maps into C. (A half plane forces constancy only for entire functions, by Liouville's theorem; here \(f\) is defined only on a disk.)
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