Let \(\displaystyle C\) be a circle, and \(\displaystyle p\) a point outside of \(\displaystyle C\). Show that if \(\displaystyle f\) is a fractional linear transformation such that \(\displaystyle f(C)=C\) and \(\displaystyle f(p)=p\), then \(\displaystyle |f'(p)|=1\).

Difficulty: 5.

Let \(\displaystyle P=\{z:\re z>0\}\) be the right half-plane \(\displaystyle f:P\to P\) regular and \(\displaystyle f(1)=1\). Prove that \(\displaystyle {|f'(1)|\le1}\).

Difficulty: 5.

For all \(\displaystyle D \subset \CC\) domain and \(\displaystyle a \in D\) there is a unique \(\displaystyle r(a, D)\) radius such that there is a conformal injection \(\displaystyle f: D \leftrightarrow S\big(0, r(a, D)\big), \ f(a)=0, \ f'(a)=1\).

Difficulty: 6.

Let \(\displaystyle F\not\subseteq G\) and \(\displaystyle D\) compex simply connected domains \(\displaystyle a\in F\), and \(\displaystyle f:F\leftrightarrow D\), \(\displaystyle g:G\leftrightarrow D\) conformal bijections such that \(\displaystyle f(a)=g(a)\). Show that \(\displaystyle |f'(a)|>|g'(a)|\).

Difficulty: 6.

Let the roots of the regular function \(\displaystyle f: S(0, 1) \to S(0, 1)\) be \(\displaystyle a_1, \dots , a_n\). Show that

\(\displaystyle |f(z)|\le \prod\limits_{i=1}^n \left|\frac{a_i -z}{1-\overline{a_i}z}\right| \qquad (|z|<1).\)

Difficulty: 6.

Let \(\displaystyle a_1,a_2,\dots\) be a sequence of complex numbers such that \(\displaystyle |a_k|<1\) and \(\displaystyle \re a_k>\frac12\) for all \(\displaystyle k\). Let

\(\displaystyle z_0=0, \qquad z_{n+1}=\frac{z_n+a_n}{1+\overline{a_n}z_n}. \)

Prove that \(\displaystyle a_n\to1\).

(IMC 2011/6 alapján)

Difficulty: 6.

Let \(\displaystyle T\), \(\displaystyle R\in \Aut\big(S(0, 1)\big)\) and \(\displaystyle T(a)=R(a)=0\). Prove that \(\displaystyle T=cR\) for some \(\displaystyle |c|=1\). Describe \(\displaystyle \Aut\big(S(0, 1)\big)\) using this observation.

Difficulty: 7.

Assume that \(\displaystyle f\) is regular on the unit disc and satisfies \(\displaystyle |f(z)|<1\). Show that

\(\displaystyle \frac{|f'(z)|}{1-|f(z)|^2} \le \frac1{1-|z|^2} . \)

Difficulty: 7.

Assume that \(\displaystyle f\in O(|z|<1)\) has image \(\displaystyle \re z>0\), and \(\displaystyle f(0)=1\). Show that

\(\displaystyle \frac{1-|z|}{1+|z|}\le |f(z)|\le \frac{1+|z|}{1-|z|}.\)

Difficulty: 7. Solution is available for this problem.

Let \(\displaystyle w: S(0, 1)\to S(0, 1)\) be regular and let \(\displaystyle |a|<1\). Show that

   a) \(\displaystyle \displaystyle \left|\frac{w(z)-w(a)}{1-\overline{w(a)}w(z)}\right|\le \left|\frac{z-a}{1-\bar{a}z}\right|\)    b) \(\displaystyle \displaystyle |w'(a)|\le\frac{1-|w(a)|^2}{1-|a|^2}\).

Difficulty: 7.

Let \(\displaystyle w:S(0, 1) \to S(0, 1)\) be regular and \(\displaystyle w(\alpha)=0\). Show that

(a) \(\displaystyle \displaystyle |w(z)|\le \left|\frac{z-\alpha}{1-\bar{\alpha}z}\right|\);       (b) \(\displaystyle |w'(a)|\le1-|\alpha|^2.\)

Difficulty: 7.

Let \(\displaystyle D=\{z\in\mathbb{C}:|z|<1\}\) be the complex unit disc and let \(\displaystyle 0<a<1\) be a real number. Suppose that \(\displaystyle f:D\to\mathbb{C}\) is a holomorphic function such that \(\displaystyle f(a)=1\) and \(\displaystyle f(-a)=-1\).

(a) Prove that

\(\displaystyle \sup_{z\in D} \big|f(z)\big| \ge \frac1a . \)

(b) Prove that if \(\displaystyle f\) has no root, then

\(\displaystyle \sup_{z\in D} \big|f(z)\big| \ge \exp \left(\frac{1-a^2}{4a}\pi\right) . \)

(Schweitzer-competition, 2012)

Difficulty: 9. Solution is available for this problem.