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    A boundary-modulus-one holomorphic map covers the unit disk

    P-CHAVAN-CA-3-5

    Area
    Complex Analysis
    Status
    draft
    Contents

    Let \(f\) be nonconstant and holomorphic on an open set containing the closed unit disk, and suppose \[ |f(z)|=1\qquad(|z|=1). \]

    1. Show that \(f\) has a zero in the open unit disk.

    2. Show that for every \(|w_0|<1\), the equation \(f(z)=w_0\) has a root in the open unit disk.

    Thus the image of \(f\) contains the unit disk.

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