Let \(f\) be entire with \(\im(f) \intersect \DD_r(z_0)\) empty. Show \(f\) must be constant without using the Casorati-Weierstrass or Picard theorems.
Solution
Write \(g(z) \da f(z) - z_0\), so \(\abs{g(z)} \geq r\). Now \(\abs{1/g(z)} \leq 1/r\) for all \(z\), so \(1/g\) is bounded. Moreover it is entire since \(f(z) \neq z_0\) for any \(z\), and so \(1/g = c\) is constant. Now unwind to get \(f(z) = z_0 + {1\over c}\), which is also constant.