A proper subdomain G of the unit disk 𝔻 is horocyclically convex (horo-convex) if, for every ω∈𝔻∩∂ G , there exists a horodisk H such that ω∈∂ H and G∩ H=∅ . In this paper we give an internal characterization of these domains, namely, that G is horo-convex if and only if any two points can be joined inside G by a C^1 curve composed with finitely many Jordan arcs with hyperbolic curvature in (-2,2) . We also give a lower bound for the hyperbolic metric of horo-convex regions as well as some of its consequences.

Let the function $$\varphi $$ be holomorphic in the unit disk $${\mathbb {D}}$$ of the complex plane $${\mathbb {C}}$$ and let $$\varphi ({\mathbb {D}})\subset {\mathbb {D}}$$ . We study the level sets and the critical points of the hyperbolic derivative of $$\varphi $$ , $$\begin{aligned} |D_{\varphi }(z)|:=\frac{(1-|z|^2)|\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$ In particular, we show how the Schwarzian derivative of $$\varphi $$ reveals the nature of the critical points.

We present the notion of lower spherical order for locally injective meromorphic functions in the unit disk, and study some properties of functions with positive lower spherical order.

Generalizing the modular and Hecke groups, we consider the subgroup Π of SL(2, ℤ[ξ]) generated by the parabolic element \(\left( {_{01}^{1\xi } } \right)\) and the elliptic element \(\left( {_{10}^{0 - 1} } \right)\), where ℤ[ξ] is the ring of polynomials in the variable ξ. For ζ ∈ ℂ and W ∈ Π, we denote by W (ζ) the matrix in SL(2, ℂ) obtained when evaluating the parameter ξ at ζ. We enumerate the elements of Π and study the relators, defined as those W ∈ ℂ for which there exists ζ ∈ ℂ with W(ζ) = ±I. Then, for W ∈ Π, we investigate the sets of ζ for which W(ζ) is not loxodromic; their union is the singular set S(Π) ⊂ ℂ. The closure of the singular set for the two-parabolic group, which is isomorphic to a free subgroup Π of index 4, has been studied extensively.

We define the hyperbolic order of any locally injective holomorphic function between arbitrary hyperbolic domains of the complex plane and study the relation between the hyperbolic order and the Schwarzian derivative for locally injective holomorphic functions from the unit disk into itself.

We discuss two-point distortion inequalities for (not necessarily normalized) univalent functions f on the unit disk D. By a two-point distortion inequality we mean an upper or lower bound on the Euclidean distance |f(a)−f(b)| in terms of dD(a,b), the hyperbolic distance between a and b, and the quantities (1−|a|2)|f′(a)|,(1−|b|2)|f′(b)|. The expression (1−|z|2)|f′(z)| measures the infinitesimal length distortion at z when f is viewed as a function from D with hyperbolic geometry to the complex plane C with Euclidean geometry. We present a brief overview of the known two-point distortion inequalities for univalent functions and obtain a new family of two-point upper bounds that refine the classical growth theorem for normalized univalent functions.

La presente ponencia explica las familias analiticas de homoformismos, conocida tambien como funcion analitica o diferencial, que puede demostrar equivalencia cuando requiere que sea solo funcion continua.

Let Φ be a Fuchsian group acting on the unit disk D. This group may be infinitely generated and may have elliptic elements. A non-constant function f meromorphic in D is called Φ-polymorphic if, for every ϕ ∈Φ there is a Moebius transformation γ such that f ∘ φ = γ ∘ f. This induces a homomorphism f* of Φ into PSL(2, ℂ). The image group Г:= f *(Φ) need not be discrete. In this paper we study function-theoretic properties of a polymorphic function f and its relation to the limit sets and fixed points of Φ and the image group Г. This has some consequences for degenerate Kleinian groups.

In 1964 Pommerenke introduced the notion of linear invariant family for locally injective analytic functions defined in the unit disk of the complex plane. Following Ma and Minda (who extended this notion to spherical geometry), we consider in this paper locally injective meromorphic functions in the unit disk. More precisely, we study families of such functions for which a certain invariant, called spherical order, is finite. Several consequences on the finiteness of the spherical order are explored, in particular the connection with the Schwarzian and normal orders, and with uniform perfectness.

An analytic function f with Schwarzian norm parallel to gf parallel to <= 2(1 + delta(2)) is shown to satisfy a pair of two-point distortion conditions, one giving a lower bound and the other an upper bound for the deviation. Conversely, each of these conditions is found to imply that parallel to gf parallel to <= 2(1 + delta(2)). Analogues of the lower bound are also developed for curves in R-n and for canonical lifts of harmonic mappings to minimal surfaces.

A subdomain G in the unit disk D is called hyperbolically convex if the non-euclidean segment between any two points in G also lies in G. We introduce the concept of constricted domain relative to the hyperbolic geometry of D and prove that a hyperbolic convex domain is constricted if and only if it is not a quasidisk. Also examples are given to illustrate these ideas.

The main new concept is the singular set S of parameter values t for which some Moebius transformation h t(x) becomes non-loxodromic. We study the structure and geometry of S and the behaviour of h t in domains \(V \subset T\backslash \bar S\), in particular the extension to values in ∂V ∩ ∂T.

Let be a Fuchsian group acting on the unit disk D. A function f meromorphic in D is polymorphic if there exists a homomorphism f of onto a group of Mobius transformations such that f = f ( ) f for 2 . A function is normal if sup 1 jzj 2 jf 0 (z)j= 1 + jf(z)j 2 < 1. First we study the behaviour of a normal polymorphic function at the xed points of and then the existence of such functions for a given type of group .

Let C(w1,w2,w3) denote the circle in Cˆ through w1,w2,w3 and let w1w2ˆ denote one of the two arcs between w1,w2 belonging to C(w1,w2,w3). We prove that a domain Ω in the Riemann sphere, with no antipodal points, is spherically convex if and only if for any w1,w2,w3∈Ω, with w1≠w2, the arc w1w2ˆ of the circle C(w1,w2,−1/w3¯) which does not contain −1/w3¯ lies in Ω. Based on this characterization we call a domain G in the unit disk D, strongly hyperbolically convex if for any w1,w2,w3∈G, with w1≠w2, the arc w1w2ˆ in D of the circle C(w1,w2,1/w3¯) is also contained in G. A number of results on conformal maps onto strongly hyperbolically convex domains are obtained.

Let ' be analytic in the unit disk D and let '(D) ‰ D; '(0) 6= 0. Then w = z='(z) has an analytic inverse z = f(w) for w 2 D, the flxed point function. This paper studies the case that '(1) = '0(1) = 1 with a growth condition for '00(x) and determines the asymptotic behaviour of various combinations of the coe-cients of ' connected with f. The results can be interpreted in various contexts of probability theory.