In this paper, we give an analogue of Jørgensen’s inequality for nonelementary groups of isometries of quaternionic hyperbolic space generated by two elements, one of which is elliptic. As an application, we obtain an analogue of Jørgensen’s inequality in the two-dimensional Möbius group of the above case.

Let S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

It is shown that if f is an analytic function of sufficiently small exponential type in the right half-plane, which takes integer values on a subset of the positive integers having positive lower density, then f is a polynomial.

An analytic function f in the unit disc belongs to F(p,q,s), if

is uniformly bounded for all a ∈ . Here is the Green function of , and φa(z)=(a−z)/(1−āz). It is shown that for 0 < γ < ∞ and |w|=1 the singular inner function exp(γ(z+w)/(z−w)) belongs to F(p,q,s), 0<s≤1, if and only if . Moreover, it is proved that, if 0<s<1, then an inner function belongs to the Möbius invariant Besov-type space for some (equivalently for all) p > max{s,1−s} if and only if it is a Blaschke product whose zero sequence {zn} satisfies .

In a scale of Fock spaces with radial weights ϕ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if ϕ(x) grows at most like (log x)2.

Let ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

For ε>0, let Σε={z∈ℂ:∣arg z∣<ε}. It has been proved (D. E. Marshall and W. Smith, Rev. Mat. Iberoamericana15 (1999), 93–116) that ∫ f−1(Σε)∣f(z)∣ dA(z)≃∫ 𝔻∣f(z)∣ dA(z) for every ε>0, uniformly for every univalent function f in the classical Bergman space A1 that fixes the origin. In this paper, we extend this result to those conformal maps on 𝔻 belonging to weighted Bergman–Orlicz classes such that f(0)=∣f′(0)∣−1=0.

We complete the investigation of growth properties of analytic functions connected with the Nevanlinna parametrization of the solutions of an indeterminate strong Hamburger moment problem.

There are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1. This result goes under the name of the de Franchis theorem. Having seen that the set of such holomorphic mappings is finite, we naturally want to obtain a bound on its cardinality. It has been known for some time that there exist various bounds depending only on the genera of the surfaces. Here we obtain ‘better’ bounds of the above type, using arguments based on the rigidity of holomorphic mappings and the hyperbolic geometry of surfaces.

We prove that if two transcendental meromorphic functions share all limit values from a set of positive linear measure on a rectifiable Jordan arc, then they share all limit values.

Functions in the meromorphic Besov, Qp and related classes are characterized in terms of double integrals of certain oscillation quantities involving chordal distances. Some of the results are analogous to the corresponding results in the analytic case.

This note contains a proof of the fact that a Jordan curve in ℝ2 with a continuous tangent line at each point admits a regular reparameterization. We extend the result both to more general curves in ℝn and to higher orders of differentiability.

Let F(z) be a rational map with degree at least three. Suppose that there exists an annulus such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.

J. W. Anderson (1996) asked whether two finitely generated Kleinian groups with the same set of axes are commensurable. We give some partial solutions.

In this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

Much research has been done on the geometry of Teichmüller space and Hamilton sequences of extremal Beltrami differentials. This paper discusses some problems concerning infinitesimal Teichmüller geodesic discs and Hamilton sequences of extremal Beltrami differentials in the tangent space of an infinite-dimensional Teichmüller space.

Using Ahlfors’ theory of covering surfaces, we prove the existence theorem for the T direction for algebroid functions dealing with multiple values which extends the results proved by Guo, Zheng and Ng and answers a question by Wang, Giao and the present authors.

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdős and Turan. More precisely, given a sequence of random polynomials, we show that, under some very general conditions, the roots tend to cluster near the unit circle, and their angles are uniformly distributed. The method we use is deterministic: in particular, we do not assume independence or equidistribution of the coefficients of the polynomial.

We prove that under a sharp growth condition meromorphic functions posses a direction such that at most four rational functions are completely ramified in any sector containing the direction.

In this paper, one model of the universal Teichmüller space is studied. By the method of construction, the lower bound of the inner radius of univalency by the Pre-Schwarzian derivative of quasidisks with infinity as an inner point (such as domains bounded by ellipses) is obtained.