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Thread: analytic self-map of unit disc

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    analytic self-map of unit disc

    $\displaystyle f(0,1) \to D(0,1)$ has a zero of order k at the origin. Show that the function is bounded in magnitude by $\displaystyle \vert z \vert ^k$.

    I want to use Schwarz Lemma because the function is analytic on the unit disc, its image is a subset of the unit disc, and f(0)=0. Then I can definitely say that $\displaystyle \vert f(z) \vert \leq \vert z \vert$, but I need more. How might I bring in powers of |z|? Would the other consequence of Schwarz Lemma ($\displaystyle \vert f'(0) \vert \leq 1$) help here?
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    consider $\displaystyle \frac{f(z)}{z^{k-1}}$?
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    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by cribby View Post
    $\displaystyle f(0,1) \to D(0,1)$ has a zero of order k at the origin. Show that the function is bounded in magnitude by $\displaystyle \vert z \vert ^k$.

    I want to use Schwarz Lemma because the function is analytic on the unit disc, its image is a subset of the unit disc, and f(0)=0. Then I can definitely say that $\displaystyle \vert f(z) \vert \leq \vert z \vert$, but I need more. How might I bring in powers of |z|? Would the other consequence of Schwarz Lemma ($\displaystyle \vert f'(0) \vert \leq 1$) help here?
    Use induction on $\displaystyle k$ : Schwarz's Lemma is the case $\displaystyle k=1$; now if the statement holds up to $\displaystyle k-1$ and $\displaystyle f$ has a zero of order $\displaystyle k$ at the origin, then by Schwarz's Lemma, $\displaystyle |f(z)|\leq |z|$, and from there $\displaystyle f(z)/z$ is a function $\displaystyle D(0,1)\rightarrow D(0,1)$ having a zero of order $\displaystyle k-1$ at the origin, i.e. $\displaystyle |f(z)/z| \leq |z|^{k-1}$.
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