The Koebe-Andreev-Thurston Circle Packing Theorem states that every triangulated planar graph has a circle-contact representation. The theorem has been generalized in various ways. The arguably most prominent generalization assures the existence of a primal-dual circle representation for every 3-connected planar graph. The aim of this note is to give a streamlined proof of this result.

In this paper, we introduce and investigate a subclass of analytic and bi-univalent functions in the open unit disk. Upper bounds for the second and third coefficients of functions in this subclass are founded. Our results, which are presented in this paper, generalize and improve those in related works of several earlier authors.

We show that for each cusp on the boundary of Maskit’s embedding M ⊂ H of the Teichmüller space of punctured tori there is a sequence of parameters in the complement ofM converging to the cusp such that the parameters correspond to discrete groups with elliptic elements. Using Tukia’s version of Marden’s isomorphism theorem we identify them as cusps on the boundary of certain deformation spaces...

Two open subsets of $${\mathbb {R}}^n$$ are called Schwartz equivalent if there exists a diffeomorphism between them that induces an isomorphism Fréchet spaces their functions. In this paper we use tools from quasiconformal geometry in order to prove the equivalence few families planar domains. We all quasidiscs equivalent. also any non-simply-connected domain whose boundary is quasicircle comp...

We prove a technical lemma, the C-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a C onedimensional mapping f : M 7→ M with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe.

Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-frac...