We verify a long-standing conjecture on the membership of univalent harmonic mappings in the Hardy space, whenever the functions have a “nice” analytic part. We also produce a coefficient estimate for these functions, which is in a sense best possible. The problem is then explored in a new direction, without the additional hypothesis. Interestingly, our ideas extend to certain classes of locally univalent harmonic mappings. Finally, we prove a Baernstein-type extremal result for the function $\log (h'+cg')$, when $f=h+\overline {g}$ is a close-to-convex harmonic function, and c is a constant. This leads to a sharp coefficient inequality for these functions.

In this article, we establish three new versions of Landau-type theorems for bounded bi-analytic functions of the form $F(z)=\bar {z}G(z)+H(z)$, where G and H are analytic in the unit disk with $G(0)=H(0)=0$ and $H'(0)=1$. In particular, two of them are sharp, while the other one either generalizes or improves the corresponding result of Abdulhadi and Hajj. As consequences, several new sharp versions of Landau-type theorems for certain subclasses of bounded biharmonic mappings are proved.

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 $e^{x+y}$ .

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.

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.

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.

A proof is given to show that a positive function on the unit disk admits a harmonic majorant if and only if it has a certain explicit upper envelope that admits a superharmonic majorant. The (logarithmic) Lipschitz regularity of this superharmonic majorant is discussed.

We study the farthest-point distance function, which measures the distance from $z\,\in \,\mathbb{C}$ to the farthest point or points of a given compact set $E$ in the plane.

The logarithm of this distance is subharmonic as a function of $z$, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure ${{\sigma }_{E}}$ has many interesting properties that reflect the topology and geometry of the compact set $E$. We prove ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for polygons inscribed in a circle, with equality if and only if $E$ is a regular $n$-gon for some odd $n$. Also we show ${{\sigma }_{E}}(E)\,=\,\frac{1}{2}$ for smooth convex sets of constant width. We conjecture ${{\sigma }_{E}}(E)\,\le \,\frac{1}{2}$ for all $E$.

Applications of minimal surface methods are made to obtain information about univalent harmonic mappings. In the case where the mapping arises as the Poisson integral of a step function, lower bounds for the number of zeros of the dilatation are obtained in terms of the geometry of the image.

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.

A harmonic function on the unit disc is of Beurling type ω if its Fourier (or Taylor) coefficients grow no faster than exp ω(|n|) as |n|→∞, where ω is a given increasing, concave function with ω(x)/x ↓ 0 as x → ∞. These harmonic functions are characterized by the growth rate of their L1-norms on circles of radius r as r → 1. The classical Schwartz result follows as a corollary by taking ω(x) = log(1+x). The Gevrey case is also included in the general result if one uses ω(x) = xα, 0 < α < 1.