9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a point zo EN such that (20) = 20 and y' (zo) = 1 then 4 is linear. [Hint: Why can one assume that zo = 0? Write y(z) = z+anz" +O(zn+¹) near 0, and prove that if k = o... (where appears k times), then Ök(z) = z+kanz" +0(zn+¹). Apply the Cauchy inequalities and let k→ ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.]
9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a point zo EN such that (20) = 20 and y' (zo) = 1 then 4 is linear. [Hint: Why can one assume that zo = 0? Write y(z) = z+anz" +O(zn+¹) near 0, and prove that if k = o... (where appears k times), then Ök(z) = z+kanz" +0(zn+¹). Apply the Cauchy inequalities and let k→ ∞ to conclude the proof. Here we use the standard O notation, where f(z) = O(g(z)) as z → 0 means that |ƒ(z)| ≤ C|g(z)| for some constant C as [z] → 0.]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 9:**
Let \(\Omega\) be a bounded open subset of \(\mathbb{C}\), and \(\varphi : \Omega \to \Omega\) a holomorphic function. Prove that if there exists a point \(z_0 \in \Omega\) such that
\[
\varphi(z_0) = z_0 \quad \text{and} \quad \varphi'(z_0) = 1
\]
then \(\varphi\) is linear.
**Hint:** Why can one assume that \(z_0 = 0\)? Write \(\varphi(z) = z + a_n z^n + O(z^{n+1})\) near 0, and prove that if \(\varphi_k = \varphi \circ \cdots \circ \varphi\) (where \(\varphi\) appears \(k\) times), then \(\varphi_k(z) = z + k a_n z^n + O(z^{n+1})\). Apply the Cauchy inequalities and let \(k \to \infty\) to conclude the proof. Here we use the standard \(O\) notation, where \(f(z) = O(g(z))\) as \(z \to 0\) means that \(|f(z)| \leq C|g(z)|\) for some constant \(C\) as \(|z| \to 0\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F62b63449-4303-4b61-8830-8254655a53b2%2Fuu0rp3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 9:**
Let \(\Omega\) be a bounded open subset of \(\mathbb{C}\), and \(\varphi : \Omega \to \Omega\) a holomorphic function. Prove that if there exists a point \(z_0 \in \Omega\) such that
\[
\varphi(z_0) = z_0 \quad \text{and} \quad \varphi'(z_0) = 1
\]
then \(\varphi\) is linear.
**Hint:** Why can one assume that \(z_0 = 0\)? Write \(\varphi(z) = z + a_n z^n + O(z^{n+1})\) near 0, and prove that if \(\varphi_k = \varphi \circ \cdots \circ \varphi\) (where \(\varphi\) appears \(k\) times), then \(\varphi_k(z) = z + k a_n z^n + O(z^{n+1})\). Apply the Cauchy inequalities and let \(k \to \infty\) to conclude the proof. Here we use the standard \(O\) notation, where \(f(z) = O(g(z))\) as \(z \to 0\) means that \(|f(z)| \leq C|g(z)|\) for some constant \(C\) as \(|z| \to 0\).
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