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Chapter 8: Problem 10
Suppose \(f(z)\) is analytic for \(|z| \leq 1\), and \(|f(z)| \geq 1\) for \(|z| \leq 1\). If \(f(0)=1\), show that \(f(z)\) is a constant.
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Evaluate the integral \(\int_{C} z /\left(\left(16-z^{2}\right)(z+i)\right) d z\), where \(C\) is the circle (a) \(|z|=2\) (b) \(|z-4|=2\) (c) \(|z+4|=2\) (d) \(|z|=\frac{1}{2}\) (e) \(|z|=5\).
If \(f(z)\) is analytic and nonzero in the disk \(\left|z-z_{0}\right| \leq r\), show that $$ \log \left|f\left(z_{0}\right)\right|=\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|f\left(z_{0}+r e^{i \theta}\right)\right| d \theta $$
Let \(f(z)\) be analytic in \(|z|
Find the maximum and minimum values of (a) \(|z(1-z)|\) on \(|z| \leq 1\) (b) \(\left|z /\left(z^{2}+9\right)\right|\) on \(1 \leq|z| \leq 2\) (c) \(\left|(3-i z)^{2}\right|\) on \(|z| \leq 1\) (d) \(\left|5+2 i z^{2}\right|\) on \(|z| \leq 1\) (e) \(\left|\frac{z-\alpha}{1-\bar{\alpha} z}\right|\) on \(|z| \leq 1\) (where \(\alpha\) with \(|\alpha|<1\) is fixed )
Set \(M(r, f)=\max _{|z|=r}|f(z)|\) and \(m(r, f)=\min _{|z|=r}|f(z)| .\) Find \(M(r, f)\) and \(m(r, f)\) for the following entire function, and indicate all points on \(|z|=r\) where the maximum and minimum occur. (a) \(f(z)=e^{z}\) (b) \(f(z)=z^{n}\) (c) \(f(z)=z^{2}+1\) (d) \(f(z)=z^{2}-z+1\)
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