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    An entire function with values in $\HH$ is constant

    E-WNTXK

    Area
    Complex Analysis
    Topics
    Conformal Maps, Entire Functions, Fractional Linear Transformations, Liouville's Theorem
    Status
    draft

    Suppose \(f\) is entire and \(f(\CC) \subseteq \HH\). Show that \(f\) must be constant.

    Solution

    Let \(T:\HH\to\DD\) be a Cayley transform, for example \[ T(w)=\frac{i-w}{i+w}. \] Then \[ F\da T\circ f:\CC\to\DD \] is entire and bounded. By Liouville’s theorem, \(F\equiv c\) for some \(c\in\DD\). Since \(T\) is injective, \[ f\equiv T^{-1}(c), \] so \(f\) is constant.

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