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Spring 2013 - Problem 10

Riemann mapping theorem

Let f ⁣:C→C\func{f}{\C}{\C} be a non-constant entire function. Without using either of the Picard theorems, show that there exist arbitrarily large complex numbers zz for which f(z)f\p{z} is a positive real.
Solution.

Suppose otherwise, and that there exists R>0R > 0 such that Re⁡f(z)≤0\Re{f\p{z}} \leq 0 for all ∣z∣≥R\abs{z} \geq R. By compactness, Re⁡f\Re{f} is bounded above by some M>0M > 0 on B(0,R)B\p{0, R} and hence all of C\C.

Notice that Ω=C∖(M,∞)\Omega = \C \setminus \p{M, \infty} is simply connected, so by the Riemann mapping theorem, there exists a conformal map φ ⁣:Ω→D\func{\phi}{\Omega}{\D}. Thus, φ∘f\phi \circ f is a bounded entire function, so by Liouville's theorem, it is a constant, i.e, φ∘f≡C\phi \circ f \equiv C. But this implies that f≡φ−1(C)f \equiv \phi^{-1}\p{C}, so ff is a constant.