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Dear Mathoverflow Community, I am looking for a reference for the following topological fact: Fact Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if yo …

This was first proved by Landau and Toeplitz in 1907. A reference for the proof (and for generalizations) is the paper Area, capacity and diameter versions of Schwarz's lemma by Burckel, Marshall, Min …

I am interested in a proof of the following fact : Suppose that $X$ is a Riemann surface homeomorphic to the Riemann sphere. Then $X$ is conformally equivalent to the Riemann sphere. Of course, this …

Hi, Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by $$\gamma(K):=\sup|f'(\infty)|,$$ where the supremum is taken o …

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex n …

I am not sure whether the following satisfies the OP's high standards for a good answer, but I thought the result was very interesting when I first learned about it a few years ago. Theorem. Let $ …

The main reference on this topic is the book "Boundary behavior of conformal maps" by Pommerenke. If the curve is $C^\infty$, then the biholomorphic mapping extends to a smooth map on the closure of t …

Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if $$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and …

Let me second Alex Eremenko's suggestion for Donald Marshall, Complex Analysis, Cambridge 2019. The proof is based on the new notion of dipole Green's function, and is especially interesting in view o …