Show that the map \(\displaystyle 1/z\) preserves cross-ratio, i.e. \(\displaystyle (\tfrac1{z_1},\tfrac1{z_2},\tfrac1{z_3},\tfrac1{z_4})=(z_1,z_2,z_3,z_4)\). Find other maps with this property.

Difficulty: 3.

What is the geometric meaning of the imaginary part of the cross ratio of four points?

Difficulty: 3.

Prove using the behavior of the function at the points \(\displaystyle 0\), \(\displaystyle \infty\) and \(\displaystyle 1\) that \(\displaystyle \re\frac{z+1}{z-1}<0\), if \(\displaystyle |z|<1\).

Difficulty: 3.

Prove using the behavior of the exponent at the points \(\displaystyle 0\), \(\displaystyle \infty\) and \(\displaystyle 1\) that

\(\displaystyle \left|e^{\frac{z+1}{z-1}}\right|<1 \quad (|z|<1). \)

Difficulty: 3.

(a) Prove that \(\displaystyle \displaystyle(z_1, z_2, z_3):= \frac{z_1 - z_3}{z_2 - z_3}\) is real if and only if \(\displaystyle z_1\), \(\displaystyle z_2\) and \(\displaystyle z_3\) are on a line.

(b) Prove that the cross-ratio \(\displaystyle \displaystyle(z_1, z_2, z_3, z_4):= \frac{z_1 - z_3}{z_2 - z_3} : \frac{z_1 - z_4}{z_2 - z_4}\) is real if and only if \(\displaystyle z_1\), \(\displaystyle z_2\), \(\displaystyle z_3\) and \(\displaystyle z_4\) are on a circline.

Difficulty: 4.

(a) Prove that all fractional linear transformations can be expressed as a composition of translations, rotations, dilations and conjugate inversion (inversion with respect to the unit circle followed by conjugation).

(b) Derive from this the basic properties of fractional linear transformations, they are bijective conformal maps of the Riemann sphere to itself that preserve the cross-ratio and circlines.

Difficulty: 4.

Prove that a map that preserves the cross-ratio is necessarily a fractional linear transfromation.

Difficulty: 5.

Show that if a map takes even one circle to a circle, then it is a fractional linear transformation.

Difficulty: 5.

(a) Prove that for all \(\displaystyle f\in\C[z]\) one can find \(\displaystyle g \in \C[z]\) with the property that \(\displaystyle g\) has no roots inside the unit disc and \(\displaystyle |g(z)|=|f(z)|\) for \(\displaystyle |z|=1\).

(b) Prove the same for meromorphic functions on \(\displaystyle \C\). For all meromorphic \(\displaystyle f\) one can find a meromorphic \(\displaystyle g\) which has no poles or zeros inside the unit disc and which satisfies \(\displaystyle |g|=|f|\) on the unit circle.

Difficulty: 5.

What are the possible poles and zeros of a fractional linear transformation that maps the unit circle to itself?

Difficulty: 5.

What are the meromorphic functions \(\displaystyle f\) that satisfy \(\displaystyle |f(z)|=1\) for \(\displaystyle |z|=1\)?

Difficulty: 5.

Prove the following statements.

(a) If \(\displaystyle T(z)\) is a fractional linear transformation, then \(\displaystyle T\) has a fixed point in \(\displaystyle \CC\cup\infty\).

(b) Given \(\displaystyle z_j\), \(\displaystyle w_j \ \ (j=1, 2, 3)\) with \(\displaystyle (z_k\ne z_j, \ w_k\ne w_j)\), then there is a unique \(\displaystyle T\) fractional linear transformation such that \(\displaystyle T(z_j)=w_j\).

(c) Describe the fractional linear transformations with \(\displaystyle 1\), \(\displaystyle 2\) or more fixed points.

Difficulty: 5.

Az \(\displaystyle f\) függvény reguláris az \(\displaystyle |z|<1+\varepsilon\) körlemezen. Igazold, hogy

\(\displaystyle \log|f(0)| \le \frac1{2\pi}\int_0^{2\pi} \log|f(e^{it})| \dt. \)

Difficulty: 5.

Show that there is exactly one conformal map which

(a) takes a given circle \(\displaystyle C\) to another circle \(\displaystyle C'\) in such a way that it takes 3 prescribed points on \(\displaystyle C\) to 3 prescribed points on \(\displaystyle C'\).

(b) takes a given circle \(\displaystyle C\) to another circle \(\displaystyle C'\) in such a way that it takes a prescribed point on \(\displaystyle C\) to a prescribed point on \(\displaystyle C'\) and a prescribed point inside \(\displaystyle C\) to a prescribed point inside \(\displaystyle C'\).

Difficulty: 5.

Let \(\displaystyle H\) be the upper half-plane. Prove that

\(\displaystyle \Aut(H)= \left\{ T(z)=\frac{az+b}{cz+d}, \ a, \ b,\ c,\ d\in \RR , \,\, \det\begin{pmatrix}a&b\\ c&d\end{pmatrix}>0 \right\} = PSL(2,\RR). \)

If an element of \(\displaystyle \Aut(H)\) is represented by a matrix \(\displaystyle \begin{pmatrix} a & b\\ c & d \end{pmatrix}\) what matrices correspond to the same map?

Difficulty: 5.

Assume that \(\displaystyle f_n\in O(D)\) and \(\displaystyle f_n \to f\ (\ne const.)\) uniformly on \(\displaystyle D\). Show that if for all \(\displaystyle n\) there is a circline \(\displaystyle K_n\) whose image under \(\displaystyle f_n\) is a circline then \(\displaystyle f\) takes all circlines to circlines.

Difficulty: 6.

If the zeros of the regular \(\displaystyle f: S(0, 1) \to S(0, 1)\) function are \(\displaystyle |a_k|<1\) complex numbers (possibly infinitely many), then

\(\displaystyle |f(0)|\le \left|\prod\limits_{i=0}^\infty a_i\right|.\)

Difficulty: 6.

What are the finite subgroups of the group of fractional linear transformations?

Difficulty: 7.

What fractional linear transformations map the right half-plane to itself?

Difficulty: 7.

Let \(\displaystyle f\) be regular on the disc \(\displaystyle |z|<1+\varepsilon\) except for finitely many poles. Assume that \(\displaystyle f(0)=1\) and that the zeros and poles of \(\displaystyle f\) inside the unit disc listed with multiplicity are \(\displaystyle \varrho_1,\varrho_2,\ldots,\varrho_n\), and \(\displaystyle p_1,p_2,\ldots,p_m\) respectively. Prove that

\(\displaystyle \frac1{2\pi}\int_{|z|=1}\log|f(z)|\cdot|dz| = \log\left|\frac{p_1p_2\ldots p_m}{ \varrho_1\varrho_2\ldots \varrho_n}\right|. \)

(If there are no zeros or poles then the respective product, that is empty, is \(\displaystyle 1\).)

Difficulty: 7.