Discussion
Let \(f : \mathbf{C} \to \mathbf{C}\) be a non-constant analytic such that \(|f(z)| \le |z|\) for \(|z| \ge 2019\). Then which of the following is false ?
Answer: A
The false statement is that \(f\) maps a line onto a circle.
The growth bound \(|f(z)| \le |z|\) for large \(|z|\) and the Cauchy estimates make \(f\) a polynomial of degree at most 1. Being non-constant, \(f(z) = az + b\) with \(a \neq 0\).
Such a map is a rotation, scaling and translation, so it sends lines to lines and circles to circles, never a line onto a circle. The other statements are true for \(az + b\): it is one-one, onto, and \(f'(z) = a \neq 0\) everywhere.
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