Additional Problems for Chapter 8
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Construct an entire function which has a zero of order $2$ at every point of the sequence $\{ \log n \}_{n \in \NN}$ and does not vanish anywhere else.
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Let $f \in \MM(\CC)$. Show that $f=g'/g$ for some $g \in \MM(\CC)$ if and only if the poles of $f$ are simple with integer residues. (Hint: For the “if” part, construct an $h \in \MM(\CC)$ with a zero or pole of order $\pm \res(f,p)$ at every pole $p$ of $f$, and with no other zeros or poles.)
-
The Mahler measure $M(P)$ of a complex polynomial map $P$ is defined as the geometric mean of $|P|$ on the unit circle:
\[
M(P) := \exp \left( \frac{1}{2\pi} \int_0^{2\pi} \log|P(e^{it})| \, dt \right).
\]
If $P(z)=C \prod_{n=1}^d (z-z_n)$, show that
\[
M(P) = |C| \prod_{n=1}^d \max \{|z_n|,1\}.
\]
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Let $a_n=1-1/n^2$. Prove that the infinite product
\[
f(z)=\prod_{n=1}^{\infty} \left( \frac{a_n-z}{1-a_n z} \right)
\]
defines a holomorphic function $f: \DD \to \DD$.
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Determine all bounded holomorphic functions $f:\HH \to \CC$ such that $f(in)=e^{-n}$ for all $n \in \NN$.
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Suppose $f \in \OO(\CC)$ satisfies $|f(z)| \leq M e^{|z|}$ for all $z$. Show that
\[
|f^{(n)}(0)| \leq M n! \left( \frac{e}{n} \right)^n \qquad n \geq 1.
\]
Conclude that $|f^{(n)}(0)|/\sqrt{n}$ stays bounded as $n \to \infty$.
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Let $f$ be a transcendental entire function with finitely many zeros. Show that $m(r):=\min_{|z|=r} |f(z)| \to 0$ as $r \to +\infty$.
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