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    Density of $f(\mathbb{H})$ for a holomorphic map of the half-plane real precisely on $\mathbb{R}$

    P-ZNVSP

    Area
    Complex Analysis
    Topics
    Open Mapping Theorem, Picard, Schwarz Reflection
    Seen at
    UGA
    Years
    2011
    Status
    draft

    Suppose \(f: \HH\union \RR\to \CC\) satisfies the following:

    • \(f(i) = i\)

    • \(f\) is continuous

    • \(f\) is analytic on \(\HH\)

    • \(f(z) \in \RR \iff z\in \RR\).

    Show that \(f(\HH)\) is a dense subset of \(\HH\).

    Solution

    Ideas:

    • If an entire function doesn’t have dense image, it’s constant by Liouville using the proof idea of Casorati-Weierstrass.

    • Conjugate \(f\) by \(T:\HH\to \DD\) where \(T(z) = {z-i\over z+i}\), then \(\tilde f(0) = 0\)

    • Use that \(T(\RR) = S^1\), so \(\abs{\tilde f(z)} = 1\) when \(\abs{z} = 1\).

    • Schwarz reflection applies to \(\tilde f\) to define an entire function – if \(f\) isn’t dense, then the extension of \(\tilde f\) isn’t dense…?

    • No clue how to use \(f(i) = i\), although it implies \(\tilde f(0) = 0\) and Schwarz applies.

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