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    A Schwarz-type bound on the upper half-plane

    E-SS8.EX-10

    Area
    Complex Analysis
    Topics
    Automorphisms, Schwarz Lemma
    Status
    draft
    1. Let \(F : \mathbb { H } \to \mathbb { C }\) be a holomorphic function that satisfies

    \[ | F (z) | \leq 1 \quad \text { and } \quad F (i) = 0. \]

    Figure 11. Successive conformal maps in Exercise 8

    Prove that

    \[ | F (z) | \leq \left| \frac {z - i}{z + i} \right| \quad \text { for all } z \in \mathbb {H}. \]

    Solution

    Let \[ \phi(z)=\frac{z-i}{z+i}. \] This maps the upper half-plane biholomorphically onto the unit disc and satisfies \(\phi(i)=0\). Define \[ g(w)=F(\phi^{-1}(w)),\qquad w\in\mathbb D. \] Then \(g\) is holomorphic on \(\mathbb D\), \(|g(w)|\le1\), and \(g(0)=F(i)=0\). By Schwarz’s lemma, \[ |g(w)|\le|w|. \] Taking \(w=\phi(z)\) gives \[ |F(z)|\le\left|\frac{z-i}{z+i}\right|, \qquad z\in\mathbb H. \]

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