Automorphism Groups and Biholomorphisms

What is the group of all biholomorphisms of \(\mathbb {D}\setminus \{0\}\)?

Let \(A_R=\{z\in \mathbb {C}\mid 1<|z|<R\}\) for \(R>1\). If \(A_R\) and \(A_{R'}\) are biholomorphic, show that \(R=R'\).

Show that if \(f:B(0,1)\to \mathbb {C}\) is holomorphic and \(\operatorname {Re}(f'(z))>0\) for all \(z\in B(0,1)\) then \(f\) is injective.

What is the group of all biholomorphisms of \(\mathbb {D}\) that fix 0? Justify your answer.

What is the group of all biholomorphisms of \(\mathbb {C}\setminus \{0\}\)?

Show that if \(D\subsetneq B(0,1)\) is simply connected and \(0\in D\) then there exists an injective, holomorphic function \(f:D\to B(0,1)\) so that \(f'(0)>1\). -Note that this is part of the proof of the Riemann mapping theorem, so please do not use the Riemann mapping theorem in the proof.

Let \(f_i:\mathbb {D}\to \mathbb {C}\) be a sequence of injective holomorphic functions so that \(f_i\to f_\infty \) uniformly on compact sets. If \(f_\infty \) is not injective, what can you say about it? Prove this.

Show that if \(f\) is an entire function and \(f\) restricted to the unit disk is a bijection, that is, \(f|_{\mathbb D}:\mathbb D\to \mathbb D\) is a bijection, then \(f\) is a rotation. That is, there exists \(c\in \mathbb C\) with \(|c|=1\) so that \(f(z)=cz\) for all \(z\).

The following statement is false: If \(\Omega \subset \mathbb C\) is any simply connected set and \(x\in \Omega \), then there exists a unique \(r\) and biholomorphism \(f:\Omega \to B(0,r)\) so that \(f(x)=0\) and \(f'(x)=1\). Fix the statement and provide a proof for the correct statement.

Note: Simply connected implies the set is open and connected.

Show that if \(f:\mathbb D\to \mathbb C\) is holomorphic and \(f'(0)\ne 0\) then there is a neighborhood of \(0\) where \(f\) is injective. Do not use the inverse function theorem.

Prove or disprove: Let \[ \Omega =\{z:\ |\operatorname {Re}(z)|<1\text { and }|\operatorname {Im}(z)|<1\}. \] If \(f:\mathbb D\to \Omega \) is a biholomorphism so that \(f(z)\) is real for all real \(z\in \mathbb D\) and \(f(0)=0\), then \[ f(iz)=i f(z). \]

Let \(a_j\in \mathbb C\) satisfy \(|a_j|\le 1\) for all \(j\ge 2\). Show that \(f:B(0,\frac {1}{100})\to \mathbb C\) defined by \[ f(z)=z+\sum _{j=2}^{\infty }a_jz^j \] is injective.

Find a biholomorphism \(\varphi \) of \(D\setminus \mathbb R^-\) onto \(D\). Here \(\mathbb R^-=\{x\in \mathbb R:x\le 0\}\).

(It will suffice to give explicitly finitely many maps whose composition is equal to \(\varphi \).)

Prove or give a counterexample: \(B_1,B_2\) are Banach spaces and \(A:B_1\to B_2\) is continuous, linear, injective, and \(A(B_1)\) is dense. Then \(A\) is a bijection.

Let \(f(z)=nz+z^n\), where \(n\) is a positive integer. Show that \(f\) is one-to-one on \(D\).

Let \(f:U\to \mathbb C\) be a holomorphic function, where \(U\) is an open set. Let \(a\in U\). A standard result in complex analysis is that if \(f'(a)\ne 0\), then there is an open neighborhood \(V\subset U\) of \(a\) on which \(f\) is injective and the image \(f(V)\) is a neighborhood of \(f(a)\). Prove this result!

Classify entire biholomorphisms. That is, what are all \(f:\mathbb C\to \mathbb C\) holomorphic with a holomorphic inverse \(g:\mathbb C\to \mathbb C\)?

Let \(f\) be an automorphism of the unit disk \(|z|<1\). Assume that \(f\) fixes two points. Prove that \(f(z)=z\) for all \(z\) in the unit disk.

Let \(U\) be an open connected domain in the complex plane, and let \(f\) be analytic on \(U\).

(a) Prove or disprove: if \(f\) is injective on \(U\), then the derivative \(df/dz\) is never zero on \(U\).

(b) Prove or disprove: if the derivative \(df/dz\) is never zero on \(U\), then \(f\) is injective on \(U\).

Consider a measurable subset \(A\) of the circle \(\{z\in \mathbb C\mid |z|=1\}\). Fix a real number \(\theta \) so that \(\theta /\pi \) is irrational. Define \[ A_\theta =\{e^{i\theta }z\mid z\in A\}, \] the rotation of \(A\) through \(\theta \) radians. Prove that if \[ A=A_\theta \] then either \(A\) has measure zero or else \(A\) is the complement of a set of measure zero. (Hint: consider the Fourier series of \(\chi _A-\chi _{A_\theta }\); here \(\chi _S\) denotes the characteristic function of \(S\).)

Let \(f\) be a holomorphic automorphism of \(\mathbb C^*\). Show that either \(f(z)=\alpha z\) or \(f(z)=\frac {\alpha }{z}\) for some complex number \(\alpha \ne 0\).

Let \(f\) be a complex valued function defined on the domain \(D\subset \mathbb C\). Suppose, that as a mapping of \(\mathbb R^2\) to \(\mathbb R^2\), \(f=u+iv\) is twice differentiable, and that the jacobian \[ \begin {pmatrix} \dfrac {\partial u}{\partial x} & \dfrac {\partial u}{\partial y}\\[0.6ex] \dfrac {\partial v}{\partial x} & \dfrac {\partial v}{\partial y} \end {pmatrix} \] is everywhere, as a linear transformation, a composition of a dilation and a rotation. Show that \(f\) is a holomorphic function.

Assume that \(f\) is a one-to-one conformal mapping of the open disc \(|z|<1\) onto a regular \(n\)-gon with center at \(0\), and \(f(0)=0\). Prove that \(f(\rho z)=\rho f(z)\) where \(\rho \) is any \(n\)-th root of \(1\). Remember that conformal maps preserve angles.

Let \(f\) be an automorphism of the unit disc \(|z|<1\). Assume that \(f\) fixes two points. Show that \(f(z)=z\) for all \(z\) in the unit disc.

Let \(f\) be an injective entire function. Put \(g(z)=f(1/z)\). Show that \(g\) does not have an essential singularity at \(0\). In particular, it follows that \(f(z)=az+b\). Why?

(a) State the definition of a Fourier transform of \(f\in L^1(\mathbb R)\);

(b) State the Fourier Inversion Theorem;

(c) Show that the Fourier transform is one-to-one on \(L^1(\mathbb R)\).

Find a biholomorphism \(\varphi \) of \(\Delta -\mathbb R^+\) onto \(\Delta \). Here \(\mathbb R^+=\{x\in \mathbb R:x\ge 0\}\). (It will suffice to give explicitly finitely many maps whose composition is equal to \(\varphi \).)

Let \(P(z)=kz+z^k\) where \(k\) is a positive integer. Show that \(P\) is one-to-one on \[ \{z:|z|<1\}. \]