Let \(f(z)\) be entire and assume values of \(f(z)\) lie outside a bounded open set \(\Omega\). Show without using Picard’s theorems that \(f(z)\) is a constant.
Under Fall 2009, question 13, the source also reprints, as parts (1) and (2), the two parts of question 1 of the same list (Parseval’s identity \(\frac{1}{2\pi}\int_0^{2\pi} \abs{f(re^{i\theta})}^2\,d\theta = \sum_{n\ge0} \abs{c_n}^2 r^{2n}\) for a power series converging in \(\abs{z} < R\), and the deduction of Liouville’s theorem). They are a separate problem, carried by the card for question 1.
The source prints this problem three times, as Fall 2009, question 13, and Fall 2011, questions 13 and 14.