Conformal Mapping

Find necessary and sufficient conditions which the complex numbers $a, b, c$, and $d$ have to satisfy so that the linear fractional transformation $z \rightarrow (a z+b) /(c z+d)$ maps the upper half plane onto itself.

In Theorem 11.14 the hypotheses were, in simplified form, that $\Omega \subset \Pi^{+}, L$ is on the real axis, and $\mathrm{Im}\left(f(z)\right) \rightarrow 0$ as $z \rightarrow L$. Use this theorem to establish analogous reflection theorems under the following hypotheses:

Suppose $R$ is a rational function such that $\left|R(z)\right| = 1$ if $\left|z\right| = 1$. Prove that

where $c$ is a constant, $m$ is an integer, and $\alpha_{1}, \ldots , \alpha_{k}$ are complex numbers such that $\alpha_{n} \neq 0$ and $\left|\alpha_{n}\right| \neq 1$. Note that each of the above factors has absolute value 1 if $\left|z\right| = 1$.

Obtain an analogous description of those rational functions which are positive on $T$. Hint: Such a function must have the same number of zeros as poles in $U$. Consider products of factors of the form

where $\left|\alpha \right| < 1$ and $\left|\beta \right| < 1$.

Suppose $f$ is a trigonometric polynomial,

and $f(\theta ) > 0$ for all real $\theta$. Prove that there is a polynomial $P(z) = c_{0}+c_{1} z+\cdots +c_{n} z^{n}$ such that

Hint: Apply Exercise 4 to the rational function $\Sigma a_{k} z^{k}$. Is the result still valid if we assume $f(\theta ) \geq 0$ instead of $f(\theta ) > 0$?

Find the fixed points of the mappings $\varphi_{\alpha }$ (Definition 12.3). Is there a straight line which $\varphi_{\alpha }$ maps to itself?

Find all complex numbers $\alpha$ for which $f_{\alpha }$ is one-to-one in $U$, where

Describe $f_{\alpha }(U)$ for all these cases.

Suppose $f(z) = z+(1 / z)$. Describe the families of ellipses and hyperbolas onto which $f$ maps circles with center at 0 and rays through 0.

Suppose $f$ and $g$ are holomorphic mappings of $U$ into $\Omega , f$ is one-to-one, $f(U) = \Omega$, and $f(0) = g(0)$. Prove that

Let $\Omega$ be the upper half of the unit disc $U$. Find the conformal mapping $f$ of $\Omega$ onto $U$ that carries $\left\{ -1,0,1\right\}$ to $\left\{ -1,-i, 1\right\}$. Find $z \in \Omega$ such that $f(z) = 0$. Find $f(i / 2)$. Hint: $f = \varphi \circ s \circ \psi$, where $\varphi$ and $\psi$ are linear fractional transformations and $s(\lambda ) = \lambda^{2}$.

Suppose $\Omega$ is a convex region, $f \in H(\Omega )$, and $\mathrm{Re}\left(f^{\prime }(z)\right) > 0$ for all $z \in \Omega$. Prove that $f$ is one-to-one in $\Omega$. Is the result changed if the hypothesis is weakened to $\mathrm{Re}\left(f^{\prime }(z)\right) \geq 0$? (Exclude the trivial case $f =$ constant). Show by an example that " convex" cannot be replaced by " simply connected".

Suppose $\Omega$ is a region, $f_{n} \in H(\Omega )$ for $n = 1,2,3, \ldots$, each $f_{n}$ is one-to-one in $\Omega$, and $f_{n} \rightarrow f$ uniformly on compact subsets of $\Omega$. Prove that $f$ is either constant or one-to-one in $\Omega$. Show that both cases can occur.

Suppose $\Omega = \left\{ x+i y:-1 < y < 1\right\} , f \in H(\Omega ),\left|f\right| < 1$, and $f(x) \rightarrow 0$ as $x \rightarrow \infty$. Prove that

and that the passage to the limit is uniform if $y$ is confined to an interval $[-\alpha , \alpha ]$, where $\alpha < 1$. Hint: Consider the sequence $\left\{ f_{n}\right\}$, where $f_{n}(z) = z+n$, in the square $\left|x\right| < 1,\left|y\right| < 1$.

What does this theorem tell about the behavior of a function $g \in H^{\infty }$ near a boundary point of $U$ at which the radial limit of $g$ exists?

Let $\mathscr {F}$ be the class of all $f \in H(U)$ such that $\mathrm{Re}\left(f\right) > 0$ and $f(0) = 1$. Show that $\mathscr {F}$ is a normal family. Can the condition " $f(0) = 1$ " be omitted? Can it be replaced by " $\left|f(0)\right| \leq 1$ "?

Let $\mathscr {F}$ be the class of all $f \in H(U)$ for which

Is this a normal family?

Suppose $\Omega$ is a region, $f_{n} \in H(\Omega )$ for $n = 1,2,3, \ldots , f_{n} \rightarrow f$ uniformly on compact subsets of $\Omega$, and $f$ is one-to-one in $\Omega$. Does it follow that to each compact $K \subset \Omega$ there corresponds an integer $N(K)$ such that $f_{n}$ is one-to-one on $K$ for all $n > N(K)$? Give proof or counterexample.

Suppose $\Omega$ is a simply connected region, $z_{0} \in \Omega$, and $f$ and $g$ are one-to-one conformal mappings of $\Omega$ onto $U$ which carry $z_{0}$ to 0. What relation exists between $f$ and $g$? Answer the same question if $f\left(z_{0}\right) = g\left(z_{0}\right) = a$, for some $a \in U$.

Find a homeomorphism of $U$ onto $U$ which cannot be extended to a continuous function on $\bar{U}$.

If $f \in \mathscr {S}$ (Definition 14.10) and $n$ is a positive integer, prove that there exists a $g \in \mathscr {S}$ such that $g^{n}(z) = f\left(z^{n}\right)$ for all $z \in U$.

Find all $f \in \mathscr {S}$ such that

Suppose $f$ is a one-to-one conformal mapping of $U$ onto a square with center at 0, and $f(0) = 0$. Prove that $f(i z) = i f(z)$. If $f(z) = \Sigma c_{n} z^{n}$, prove that $c_{n} = 0$ unless $n-1$ is a multiple of 4. Generalize this: Replace the square by other simply connected regions with rotational symmetry.

Let $\Omega$ be a bounded region whose boundary consists of two nonintersecting circles. Prove that there is a one-to-one conformal mapping of $\Omega$ onto an annulus. (This is true for every region $\Omega$ such that $S^{2}-\Omega$ has exactly two components, each of which contains more than one point, but this general situation is harder to handle).

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.

Here is yet another proof of Theorem 14.22. If $f$ is as in 14.22, repeated use of the reflection principle extends $f$ to an entire function such that $\left|f(z)\right| = 1$ whenever $\left|z\right| = 1$. This implies $f(z) = \alpha z^{n}$, where $\left|\alpha \right| = 1$ and $n$ is an integer. Complete the details.

Iteration of Step 2 in the proof of Theorem 14.8 leads to a proof (due to Koebe) of the Riemann mapping theorem which is constructive in the sense that it makes no appeal to the theory of normal families and so does not depend on the existence of some unspecified subsequence. For the final step of the proof it is convenient to assume that $\Omega$ has property $(h)$ of Theorem 13.11. Then any region conformally equivalent to $\Omega$ will satisfy $(h)$. Recall also that $(h)$ implies $(j)$, trivially.

By Step 1 in Theorem 14.8 we may assume, without loss of generality, that $0 \in \Omega , \Omega \subset U$, and $\Omega \neq U$. Put $\Omega = \Omega_{0}$. The proof consists in the construction of regions $\Omega_{1}, \Omega_{2}, \Omega_{3}, \ldots$ and of functions $f_{1}, f_{2}, f_{3}, \ldots$, so that $f_{n}\left(\Omega_{n-1}\right) = \Omega_{n}$ and so that the functions $f_{n} \circ f_{n-1} \circ \cdots \circ f_{2} \circ f_{1}$ converge to a conformal mapping of $\Omega$ onto $U$.

Complete the details in the following outline.

Prove that $\sum_{n = 1}^{\infty }\left(1-r_{n}\right)^{2} < \infty$, where $\left\{ r_{n}\right\}$ is the sequence which occurs in Exercise 26. Hint:

Suppose that in Exercise 26 we choose $\alpha_{n} \in U-\Omega_{n-1}$ without insisting that $\left|\alpha_{n}\right| = r_{n}$. For example, insist only that

Will the resulting sequence $\left\{ \psi_{n}\right\}$ still converge to the desired mapping function?

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

Let $\Lambda$ be the set of all linear fractional transformations. If $\left\{ \alpha , \beta , \gamma , \delta \right\}$ is an ordered quadruple of distinct complex numbers, its cross ratio is defined to be

If one of these numbers is $\infty$, the definition is modified in the obvious way, by continuity. The same applies if $\alpha$ coincides with $\beta$ or $\gamma$ or $\delta$.

For $z \in \bar{U}, z^{2} \neq 1$, define

choosing the branch of $\log$ that has $\log 1 = 0$. Describe $f(E)$ if $E$ is

If $\varphi_{\alpha }$ is as in Definition 12.3, show that