Li et al. [A spectral radius type formula for approximation numbers of composition operators, J. Funct. Anal. 267(12) (2014), 4753-4774] proved a spectral radius type formula for the approximation numbers of composition operators on analytic Hilbert spaces with radial weights and on $H^{p}$ spaces, $p\geq 1$, involving Green capacity. We prove that their formula holds for a wide class of Banach spaces of analytic functions and weights.

Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$ . Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$ .

We prove by methods of harmonic analysis a result on the existence of solutions for twisted cohomological equations on translation surfaces with loss of derivatives at most $3+$ in Sobolev spaces. As a consequence we prove that product translation flows on (three-dimensional) translation manifolds which are products of a (higher-genus) translation surface with a (flat) circle are stable in the sense of A. Katok. In turn, our result on product flows implies a stability result of time- $\tau $ maps of translation flows on translation surfaces.

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

We show that condenser capacity varies continuously under holomorphic motions, and the corresponding family of the equilibrium measures of the condensers is continuous with respect to the weak-star convergence. We also study the behavior of uniformly perfect sets under holomorphic motions.

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

In this paper we study a class ${\mathcal{Z}}_{H}$ of harmonic mappings on the open unit disk $\mathbb{D}$ in the complex plane that is an extension of the classical (analytic) Zygmund space. We extend to the elements of this class a characterisation that is valid in the analytic case. We also provide a similar result for a closed separable subspace of ${\mathcal{Z}}_{H}$ which we call the little harmonic Zygmund space.

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

Assume that $\unicode[STIX]{x1D6FA}$ and $D$ are two domains with compact smooth boundaries in the extended complex plane $\overline{\mathbf{C}}$. We prove that every quasiconformal mapping between $\unicode[STIX]{x1D6FA}$ and $D$ mapping $\infty$ onto itself is bi-Lipschitz continuous with respect to both the Euclidean and Riemannian metrics.

Let $D\subset \mathbb{C}$ be a domain with $0\in D$. For $R>0$, let $\widehat{\unicode[STIX]{x1D714}}_{D}(R)$ denote the harmonic measure of $D\cap \{|z|=R\}$ at $0$ with respect to the domain $D\cap \{|z|<R\}$ and let $\unicode[STIX]{x1D714}_{D}(R)$ denote the harmonic measure of $\unicode[STIX]{x2202}D\cap \{|z|\geqslant R\}$ at $0$ with respect to $D$. The behavior of the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions $\unicode[STIX]{x1D714}_{D}$ and $\widehat{\unicode[STIX]{x1D714}}_{D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, $\unicode[STIX]{x1D714}_{D}(R)\leqslant \widehat{\unicode[STIX]{x1D714}}_{D}(R)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0\in D$ and all $R>0$,

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{D}(R)\geqslant C\widehat{\unicode[STIX]{x1D714}}_{D}(R)?\end{eqnarray}$$

$D$

We establish a boundary Schwarz lemma for solutions to nonhomogeneous biharmonic equations.

We give an equality condition for a symmetrization inequality for condensers proved by F.W. Gehring regarding elliptic areas. We then use this to obtain a monotonicity result involving the elliptic area of the image of a holomorphic function f.

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.

While the existence of conformal mappings between doubly connected domains is characterized by their conformal moduli, no such characterization is available for harmonic diffeomorphisms. Intuitively, one expects their existence if the domain is not too thick compared to the codomain. We make this intuition precise by showing that for a Dini-smooth doubly connected domain Ω* there exists a ε > 0 such that for every doubly connected domain Ω with ModΩ* < ModΩ < ModΩ* + ε there exists a harmonic diffeomorphism from Ω onto Ω*.

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.