A topological dichotomy with applications to complex analysis
Let $X$ be a compact topological space, and let $D$ be a subset of $X$. Let $Y$ be a Hausdorff topological space. Let $f$ be a continuous map of the closure of $D$ to $Y$ such that $f(D)$ is open. Let $E$ be any connected subset of the complement (to $Y$) of the image $f(\partial D)$ of the boundary $\partial D$ of $D$. Then $f(D)$ either contains $E$ or is contained in the complement of $E$.
Applications of this dichotomy principle are given, in particular for holomorphic maps, including maximum and minimum modulus principles, an inverse boundary correspondence, and a proof of Haagerup's inequality for the absolute power moments of linear combinations of independent Rademacher random variables. (A three-line proof of the main theorem of algebra is also given.) More generally, the dichotomy principle is naturally applicable to conformal and quasiconformal mappings.