Let $p(z)= z{f}^{\prime } (z)/ f(z)$ for a function $f(z)$ analytic on the unit disc $\mid z\mid \lt 1$ in the complex plane and normalised by $f(0)= 0, {f}^{\prime } (0)= 1$. We provide lower and upper bounds for the best constants ${\delta }_{0} $ and ${\delta }_{1} $ such that the conditions ${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\mid \lt {e}^{{\delta }_{0} / 2} $ and $\mid p(w)/ p(z)\mid \lt {e}^{{\delta }_{1} } $ for $\mid z\mid , \mid w\mid \lt 1$ respectively imply univalence of $f$ on the unit disc.

Conformal slit maps play a fundamental theoretical role in analytic function theory and potential theory. A lesser-known fact is that they also have a key role to play in applied mathematics. This review article discusses several canonical conformal slit maps for multiply connected domains and gives explicit formulae for them in terms of a classical special function known as the Schottky–Klein prime function associated with a circular preimage domain. It is shown, by a series of examples, that these slit mapping functions can be used as basic building blocks to construct more complicated functions relevant to a variety of applied mathematical problems.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

Our main aim is to investigate the properties of harmonic ν-Bloch mappings. Firstly, we establish coefficient estimates and a Landau theorem for harmonic ν-Bloch mappings, which are generalizations of the corresponding results in Bonk et al. [‘Distortion theorems for Bloch functions’, Pacific. J. Math.179 (1997), 241–262] and Chen et al. [‘Bloch constants for planar harmonic mappings’, Proc. Amer. Math. Soc.128 (2000), 3231–3240]. Secondly, we obtain an improved Landau theorem for bounded harmonic mappings. Finally, we obtain a Marden constant for harmonic mappings.

In this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

We investigate several inclusion relationships and other interesting properties of certain subclasses of p-valent meromorphic functions, which are defined by using a certain linear operator, involving the generalized multiplier transformations.

The main result shows that a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighbourhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro–Warschawski–Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp.

Let f be a polynomial of degree n≥2 with f(0)=0 and f′(0)=1. We prove that there is a critical point ζ of f with ∣f(ζ)/ζ∣≤1/2 provided that the critical points of f lie in the sector {reiθ:r>0,∣θ∣≤π/6}, and ∣f(ζ)/ζ∣<2/3 if they lie in the union of the two rays {1+re±iθ:r≥0}, where 0<θ≤π/2.

We study some classes of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small in 1984. The first section reviews the basic concepts and describes the shear construction. The main body of the paper deals with the geometry of the classes constructed.

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.

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.

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.

An extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature (), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

In this paper, we shall show that the constant in Smale's mean value theorem can be reduced to two for a large class of polynomials which includes the odd polynomials with nonzero linear term.

A function is called strongly unbounded on a domain D if there exists a sequence in D on which f and all its derivatives tend to infinity. A result of Gordon is generalized to show that an unbounded analytic function on a quasidisk is always strongly unbounded there.

Extremal partitions of domains into configurations of certain to pological form are studied. The extremal value of the weighted sum of reduced moduli of circular domains and digons is obtained. These results are applied to some problems about distortion under bounded conformal maps of the unit disk with two preassigned values.

We investigate the location and separation of zeros of certain three-term linear combination of translates of polynomials. In particular, we find an interval of the form I = [−1, 1 + h], h > 0 such that for a polynomial f, all of whose zeros are real, and λ ∈ I, all zeros of f (x + 2ic) + 2λf (x) + f (x – 2ic) are also real.

We develop sharp conditions for various types of starlikeness for functions analytic in the unit disk with bounded derivatives. We also describe the precise range {zf′(z)/f(z): z ∈ D, f ∈ }, where f ∈ means f (0) = 0, f′(0) = 1, and |f′(z) - 1 |< ≦ λ in the unit disc D, and draw some cnoslusions from that.

We study an important subclass of quasicircles, namely, symmetric quasicircles. Several characterizations for quasicircles, such as the reverse triangle inequality, the M -condition and the quasiconformal extension property, have been extended to symmetric quasicircles by Becker and Pommerenke and by Gardiner and Sullivan. In this paper we establish several relations among various domain constants such as quasiextremal distance constants, (local) reflection constants and (local) extension constants for this class. We also give several characterizations for symmetric quasicircles such as the strong quadrilateral inequality and the strong extremal distance property. They correspond to the quadrilateral inequality and the extremal distance property for quasicircles.

Let F(z) be a continuous complex-valued function defined on the closed upper half plane H whose generalized derivative ∂F(z) is unbounded. In this paper, we discuss the relationship between the increasing order of ]∂F(x + iy)] when y → 0 and that of λf(x, t) ](F(x + t) − 2F(x) + F(x − t))/t], (x, t ∈ R), when t → 0.