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    Maximum modulus principle for open mappings

    P-CHAVAN-CA-5-3

    Area
    Complex Analysis
    Status
    draft
    Contents

    Let \(f:\Omega\to\mathbb C\) be an open mapping on an open set \(\Omega\subseteq\mathbb C\). Define \(|f|(z)=|f(z)|\).

    1. Show that \(|f|\) cannot have a local maximum at a point of \(\Omega\).

    2. If \(\overline\Omega\) is compact and \(f\) is continuous on \(\overline\Omega\), show that \(|f|\) attains its maximum on \(\partial\Omega\).

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