The main aim of this article is to establish analogues of Landau’s theorem for solutions to the $\overline{\unicode[STIX]{x2202}}$ -equation in Dirichlet-type spaces.

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying

$$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$

$\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$

${\mathcal{F}}_{\unicode[STIX]{x1D706}}$

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

In this paper, we study quasiconformal extensions of harmonic mappings. Utilizing a complex parameter, we build a bridge between the quasiconformal extension theorem for locally analytic functions given by Ahlfors [‘Sufficient conditions for quasiconformal extension’, Ann. of Math. Stud.79 (1974), 23–29] and the one for harmonic mappings recently given by Hernández and Martín [‘Quasiconformal extension of harmonic mappings in the plane’, Ann. Acad. Sci. Fenn. Math.38 (2) (2013), 617–630]. We also give a quasiconformal extension of a harmonic Teichmüller mapping, whose maximal dilatation estimate is asymptotically sharp.

In this paper, for the convolution and convex combination of harmonic mappings, the radii of univalence, full convexity and starlikeness of order $\unicode[STIX]{x1D6FC}$ are explored. All results are sharp. By way of application, the univalent radius and the Bloch constant of the convolution of two bounded harmonic mappings are obtained.

We discuss the existence of finite critical trajectories connecting two zeros in certain families of quadratic differentials. In addition, we reprove some results about the support of the limiting root-counting measures of the generalised Laguerre and Jacobi polynomials with varying parameters.

Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25] gave a simple and useful univalence criterion for harmonic functions, usually called the shear construction. However, the application of this theorem is limited to planar harmonic mappings that are convex in the horizontal direction. In this paper, a natural generalisation of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic function is a union of two sets that are convex in the horizontal direction.

For holomorphic functions $f$ in the unit disk $ \mathbb{D} $ with $f(0)= 0$, we prove a modulus growth bound involving the logarithmic capacity (transfinite diameter) of the image. We show that the pertinent extremal functions map the unit disk conformally onto the interior of an ellipse. We prove a modulus growth bound for elliptically schlicht functions in terms of the elliptic capacity ${\mathrm{d} }_{\mathrm{e} } f( \mathbb{D} )$ of the image. We also show that the function ${\mathrm{d} }_{\mathrm{e} } f(r \mathbb{D} )/ r$ is increasing for $0\lt r\lt 1$.

For functions u, subharmonic in the plane, let

and let N(r,u) be the integrated counting function. Suppose that is a non-negative non-decreasing convex function of log r for which for all small r and , where 1 < ρ < 2, and define

A sharp upper bound is obtained for and a sharp lower bound is obtained for .

This paper presents a sharp boundary growth estimate for all positive superharmonic functions u in a smooth domain Ω in ℝ2 satisfying the nonlinear inequality where c>0, α∈ℝ and p>0, and δΩ(x) stands for the distance from a point x to the boundary of Ω. A result is applied to show the existence of nontangential limits of such superharmonic functions.

We introduce the class of Harnack domains in which a Harnack type inequality holds for positive harmonic functions with bounds given in terms of the distance to the domain's boundary. We give conditions connecting Harnack domains with several different complete metrics. We characterize the simply connected plane domains which are Harnack and discuss associated topics. We extend classical results to Harnack domains and give applications concerning the rate of growth of various functions defined in Harnack domains. We present a perhaps new characterization for quasidisks.

If Ω is a ring region with starlike boundary components α and β, then we show for each λ > 0 there exists a ring region ω ⊂ Ω with ∂ ω = α ∪ ϒ, α ∩ ϒ = φ such that there is a harmonic function V in ω satisfying (a) V(z) = 0 for z ∈ α, (b) V(z) = 1 for z ∈ ϒ, (c) | grad V(z)| = λ for z ∈ ϒ ∩ Ω. Furthermore, we show when ω is not equal to Ω; that is, that is, there is a non-trival solution.

Given a measurable function k non-negative a.e. on the circle |z| = 1, when is the outer function Tk (see(1.3)) continuous on the disk |z| < 1 and further, Dirichlet-finite: We shall show, among other results, that the answer is in the positive if , with ess inf k > 0.

In a paper of the same title [3] Ch. Pommerenke and the author proved several results concerning the distances of Fekete points. In the present paper I will show that the same methods can be adapted to give an answer to a problem which we could not solve at the time.

Let E be a continuum and n ≥ 4 a given positive integer. A system of points z1, …, zn ∈ E that maximizes

is called a system of Fekete points. Such a system may not be unique.