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    Schwarz lemma for functions with positive real part

    P-TIE-F15-15

    Area
    Complex Analysis
    Status
    draft
    Contents
    1. Let \(f ( z ) \in H ( \mathbb { D } ) , \operatorname { R e } ( f ( z ) ) > 0 , f ( 0 ) = a > 0\) . Show that

    \[ | \frac { f ( z ) - a } { f ( z ) + a } | \leq | z | , | f ^ { \prime } ( 0 ) | \leq 2 a . \]

    1. Show that the above is still true if \(\mathrm { R e } ( f ( z ) ) > 0\) is replaced with \(\mathrm { R e } ( f ( z ) ) \geq 0\)

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