Additional Problems for Chapter 4
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Fix $n \in \NN$. Find all holomorphic functions $f: \DD \to \DD$ which satisfy the conditions
\[
\begin{aligned}
f^{(k)}(0) & =0 \quad \text{for} \quad 0 \leq k \leq n-1 \\
|f^{(n)}(0)| & =n!
\end{aligned}
\]
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Suppose $f: \DD \to \DD$ is holomorphic and $f(0)=i/40$. At most how many zeros can $f$ have in the disk $\DD(0,1/2)$? Show that your answer is optimal. (Hint: Use the fact that a holomorphic function $f:\DD \to \DD$ is dominated by any finite Blaschke product whose zeros are among zeros of $f$; see Problem 17 of Chapter 4.)
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Suppose $f \in \OO(\DD)$ satisfies $|f| \leq 1$ everywhere in $\DD$. Assume $f(r) \to 1$ and $f'(r) \to 0$ as $r \in [0,1)$ tends to $1$. Prove that $f=1$ everywhere in $\DD$. (Hint: Use the Schwarz lemma to show that $f(\DD) \subset \DD$ leads to a contradiction.)
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Let $\TT_r := \{ z \in \CC : |z|=r \}$. If $f$ is a non-constant holomorphic function in a neighborhood of the closed unit disk, show that
\[
\diam(f(\TT_r)) \leq r \diam(f(\TT_1))
\]
for every $0 < r < 1$.
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Suppose $f \in \OO(\DD)$ and $\myre(f(z))>0$ for all $z \in \DD$. Show that $|f'(0)| \leq 2 \myre(f(0))$.
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Suppose $f\in \OO(\DD)$, $f(0)=1$, and $\myre(f(z))>0$ for all $z \in \DD$. Show that for all $z \in \DD$,
\[
\frac{1-|z|}{1+|z|} \leq |f(z)| \leq \frac{1+|z|}{1-|z|}
\]
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Suppose $P$ is a complex polynomial of degree $d \geq 1$ whose roots all belong to $\DD$. Let $P^{\ast}(z) := z^d \ov{P(1/\bar{z})}$. Prove that all roots of the polynomial $P(z) + P^{\ast}(z)$ belong to the unit circle. (Hint: Use $P, P^{\ast}$ to form a suitable Blaschke product.)
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Let $p_1, \ldots, p_n \in \DD$. If the polynomial $f(z)=(z-p_1)\cdots(z-p_n)$ satisfies $|f(z)|\leq 1$ for $|z|=1$, show that $p_1=\cdots=p_n=0$.
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Let $p \in \DD$ and, as usual, let $\varphi_p \in \Aut(\DD)$ be defined by $\varphi_p(z)=(z-p)/(1-\ov{p}z)$. Show that for $|z|=1$ the three points $z,p,-\varphi_p(z)$ are co-linear. This gives a visual representation of the action of $\varphi_p$ on the unit circle in terms of the Euclidean geometry of the disk.
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Prove that $f \in \Aut(\Chat)$ is an involution if and only if $\tau(f)=0$.
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Let $U_\ve = \DD(1,\ve)$. Show that for every $\ve>0$ there is an $f \in \Aut(\DD)$ such that $f(\DD \sm U_\ve) \subset U_\ve$. (Hint: Move the problem to the upper half-plane $\HH$, replacing $1 \in \bd \DD$ with $\infty \in \bd \HH$.)
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Suppose $U,V \subset \DD$ are open and $f: U \to V$ is a biholomorphism which preserves the hyperbolic metric of $\DD$, that is, $|f'(z)|/(1-|f(z)|^2)=1/(1-|z|^2)$ for all $z \in U$. Show that $f$ extends uniquely to an element of $\Aut(\DD)$. (Hint: After pre- and post-composing with disk automorphisms, we may assume $U,V$ are neighborhoods of $0$, with $f(0)=0, f'(0)=1$. Show that $f$ is the identity map near $0$.)
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Show that the hyperbolic geodesics in $\DD$ are the stereographic projections of the intersections of vertical planes $ax_1+bx_2+c=0$ in $\RR^3$ with the southern hemisphere $\{ (x_1,x_2,x_3) \in {\mathbb S}^2: x_3<0 \}$.
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For any $p, z \in \DD$, let $B(z)$ be the hyperbolic ball centered at $z$ of radius $\dist_{\DD}(z,p)$.
Show that if we fix $p$ but let $z \to q \in \bd \DD$, the ball $B(z)$ converges to a Euclidean disk $B \subset \DD$ which is tangent to $\bd \DD$ at $q$ and has $p$ on its boundary. Such a $B$ is called a horoball at $q$. Prove the
following statements:
(i) Every $f \in \Aut(\DD)$ maps horoballs at $q$ to horoballs at $f(q)$.
(ii) A non-identity $f \in \Aut(\DD)$ preserves some horoball at $q$ if and only if $f$ is parabolic with the unique fixed point $q$.
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Show that the upper half-plane $\HH$ can be identified with the subgroup of $\Aut(\HH)$ consisting of real affine maps $z \mapsto az+b$ with $a>0, b\in \RR$. This gives $\HH$ the structure of a (Lie) group with the identity element $i$. Verify that left multiplications $z \mapsto gz$ in this group are conformal automorphisms, but the same is not true for right multiplications $z \mapsto zg$.
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Show that any pair of unit tangent vectors in the hyperbolic disk $\DD$ can be mapped to one another by a unique automorphism. More precisely, given $p,q \in \DD$ and tangent vectors $v \in T_p\DD, w \in T_q\DD$ of hyperbolic length $1$ there is a unique $f \in \Aut(\DD)$ such that $f(p)=q$ and $f'(p)v = w$. Conclude that $\Aut(\DD)$ can be naturally identified with the unit tangent bundle of $\DD$ consisting of all pairs $(p,v)$ where $p \in \DD$ and $v \in T_p\DD$ with $\| v \|_{\DD} =1$.
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Both $\Aut(\Chat) \cong \PSL_2(\CC)$ and $\Aut(\HH) \cong \PSL_2(\RR)$ are simple groups, i.e., they carry no non-trivial proper normal subgroups. This is equivalent to the statement that the group $\SL_2(\F)$ of $2\times 2$ matrices with determinant $1$ over the field $\F=\CC$ or $\RR$ contains no proper normal subgroup larger than its center $\{ \pm I \}$. Complete the following outline of the proof for the case $\F=\RR$ (the case $\F=\CC$ is almost identical):
(i) $\SL_2(\RR)$ is generated by the parabolic elements of the form $T_x=\begin{bmatrix} 1 & x \\ 0 & 1 \end{bmatrix}$ and $S_x=\begin{bmatrix} 1 & 0 \\ x & 1 \end{bmatrix}$ with $x \neq 0$.
(ii) If $N$ is a normal subgroup of $\SL_2(\RR)$ larger than $\{ \pm I \}$, then it must contain the hyperbolic element $D=\begin{bmatrix} a & 0 \\ 0 & a^{-1} \end{bmatrix}$ for some $a \neq \pm 1$. It suffices to check that $N$ contains some hyperbolic element, for then a conjugation in $\SL_2(\RR)$ will put that element in the form $D$. Take $A \in N$, $A \neq \pm I$. If $A$ is parabolic, after a conjugation we may assume $A=T_x$ for some $x \neq 0$. Then the commutator $AS_1A^{-1}S_1^{-1} \in N$ is hyperbolic. If $A$ is elliptic, after a conjugation we may assume $A=\begin{bmatrix} \ \ \ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$. Then the commutator $AT_1A^{-1}T_1^{-1} \in N$ is hyperbolic.
(iii) If $N$ is a normal subgroup of $\SL_2(\RR)$ containing $D=\begin{bmatrix} a & 0 \\ 0 & a^{-1} \end{bmatrix}$ for some $a \neq \pm 1$, then $N$ contains all commutators $DT_xD^{-1}T_x^{-1}=T_{(a^2-1)x}$ and $DS_xD^{-1}S_x^{-1}=S_{(a^{-2}-1)x}$, and therefore it contains $T_x, S_x$ for all $x \in \RR$. Hence, $N=\SL_2(\RR)$.
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