Iteration of Holomorphic Maps

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Problem 5.2.22 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.2 "Geometry of Analytic Functions"

(a)(b)

Let $f(z) = \dfrac {z}{(1+z)^2}$ . Let $V = \left\{ z \in \mathbb {C} : 0 < \left|z\right| < \frac12, \frac{3}{4}\pi < \arg z < \frac54\pi \right\}$ .

(a)

Show that if $z \in V$ , then $\left|f(z)\right| < \left|z\right|$ and $f(z) \in V$ .

(b)

Prove that if $z \in V$ , then $\lim_{n \to \infty } f^{(n)}(z) = 0$ , where $f^{(n)}(z) = f \circ f \circ \cdots \circ f$ is the $n$ -fold composition of $f$ .

Problem 5.5.4 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.5 "Series and Normal Families"

Suppose $f$ is analytic in $U = \left\{ \left|z\right|<1\right\}$ with $f(0)=0$ and $\left|f(z)\right|<1$ for all $z \in U$ . If the sequence $\left\{ f_n\right\}$ is defined by composition

f_n(z) = \underbrace{f(f(\cdots f(z))\cdots )}_{n}

and

f_n(z) \to g(z)

for all $z \in U$ , prove that either $g(z) \equiv 0$ or $g(z) \equiv z$ .

Problem 5.5.5 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.5 "Series and Normal Families"

(a)(b)

Let $\left\{ f_n\right\}$ be a sequence of analytic functions in a domain $D$ which converges uniformly on compact subsets of $D$ to a function $f$ on $D$ .

(a)

Prove that if $f_n(z) \neq 0$ for all $n\geq 1$ and $z \in D$ , then either $f$ is identically zero in $D$ or $f(z)\neq 0$ for all $z \in D$ .

(b)

If each $f_n$ is one-to-one on $D$ , show that $f$ is either constant or one-to-one on $D$ .

Problem 5.5.6 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.5 "Series and Normal Families"

Let $D \subset \mathbb {C}$ be a bounded domain, and let $\left\{ f_n\right\}$ be a sequence of analytic automorphisms of $D$ such that

\lim _{n\to \infty } f_n(a) = b \in \partial D

for some point $a \in D$ . Prove that

\lim _{n\to \infty } f_n(z) = b

for every $z \in D$ .

Problem 4.96 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Section 3 "Existence of the Mapping Function"

(Continuation of problems 93, 94.) The limit $\lim \limits_{n \to \infty } f_{n}(z) = f(z)$ exists at every point $z$ of $\mathfrak {R}$ and the convergence is uniform in every interior domain of $\mathfrak {R}$ . The limit function $f(z)$ maps $\mathfrak {R}$ schlicht onto the interior of the unit circle.

Remark.

See Problem 258 of Part III.

Problem 2.14.24 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.14 "Conformal Mapping"

Complete the details in the following proof of Theorem 14.22. Suppose $1 < R_{2} < R_{1}$ and $f$ is a one-to-one conformal mapping of $A\left(1, R_{1}\right)$ onto $A\left(1, R_{2}\right)$ . Define $f_{1} = f$ and $f_{n} = f \circ f_{n-1}$ . Then a subsequence of $\left\{ f_{n}\right\}$ converges uniformly on compact subsets of $A\left(1, R_{1}\right)$ to a function $g$ . Show that the range of $g$ cannot contain any nonempty open set (by the three-circle theorem, for instance). On the other hand, show that $g$ cannot be constant on the circle $\left\{ z:\left|z\right|^{2} = R_{1}\right\}$ . Hence $f$ cannot exist.

Problem 2.14.29 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.14 "Conformal Mapping"

(a)(b)(c)

Suppose $\Omega$ is a bounded region, $a \in \Omega , f \in H(\Omega ), f(\Omega ) \subset \Omega$ , and $f(a) = a$ .

(a)

Put $f_{1} = f$ and $f_{n} = f \circ f_{n-1}$ , compute $f_{n}^{\prime }(a)$ , and conclude that $\left|f^{\prime }(a)\right| \leq 1$ .

(b)

If $f^{\prime }(a) = 1$ , prove that $f(z) = z$ for all $z \in \Omega$ . Hint: If

f(z) = z+c_{m}(z-a)^{m}+\cdots

compute the coefficient of $(z-a)^{m}$ in the expansion of $f_{n}(z)$ .

(c)

If $\left|f^{\prime }(a)\right| = 1$ , prove that $f$ is one-to-one and that $f(\Omega ) = \Omega$ .

Hint: If $\gamma = f^{\prime }(a)$ , there are integers $n_{k} \rightarrow \infty$ such that $\gamma^{n_{k}} \rightarrow 1$ and $f_{n_{k}} \rightarrow g$ . Then $g^{\prime }(a) = 1$ , $g(\Omega ) \subset \Omega$ (by Exercise 20, Chap. 10), hence $g(z) = z$ , by part (b). Use $g$ to draw the desired conclusions about $f$ .

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.