There is a universal constant $0<r_0<1$ with the following property. Suppose that $f$ is an analytic function on the unit disk $\D$, and suppose that there exists a constant $M>0$ so that the Euclidean area, counting multiplicity, of the portion of $f(\D)$ which lies over the disk $D(f(0),M)$, centered at $f(0)$ and of radius $M$, is strictly less than the area of $D(f(0),M)$. Then $f$ must send $r_0\bar{\D}$ into $D(f(0),M)$. This answers a conjecture of Don Marshall.
The main aim of this article is to establish certain relationships between $K$-quasiconformal harmonic mappings and John disks. The results of this article are the generalizations of the corresponding results of Ch.~Pommerenke \cite{Po}.
We consider quasiconformal mappings of the unit disk that have a planar extension which have $p$-integrable distortion. In this paper, we establish a bound for the modulus of continuity for the inverse mapping and show sharpness of this bound. Furthermore, we also obtain bounds for the compression multifractal spectra of such mappings.
In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of $K$-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investigate the linear measure distortion and the Lipschitz continuity on $K$-quasiconformal harmonic mappings of the unit disk onto a radial John disk. Finally, using Pommerenke interior domains, we characterize certain differential properties of $K$-qua…
We prove some sharp inequalities for complex harmonic functions on the unit disk. The results extend a M. Riesz conjugate function theorem and some well-known estimates for holomorphic functions. We apply some of results to the isoperimetric inequality for harmonic mappings.
We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.
In this paper, we prove a Second Main Theorem for holomorphic mappings in a disk whose image intersects some families of nonlinear hypersurfaces (totally geodesic hypersurfaces with respect to a meromorphic connection) in the complex projective space $\mathbb{P}^k$. This is a generalization of Cartan's Second Main Theorem. As a consequence, we establish a uniqueness theorem for holomorphic mappings which intersects $O(k^3)$ many totally geodesic hypersurfaces.
The conformal mapping $f(z)=(z+1)^2 $ from $\mathbb{D}$ onto the standard cardioid has a homeomorphic extension of finite distortion to entire $\mathbb{R}^2 .$ We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $\mathbb{D}$ onto cardioid-type domains.
For a univalent smooth mapping $f$ of the unit disk $\ID$ of complex plane onto the manifold $f(\ID)$, let $d_f(z_0)$ be the radius of the largest univalent disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). The main aim of the present article is to investigate how the radius $d_h(z_0)$ varies when the analytic function $h$ is replaced by a sense-preserving harmonic function $f=h+\overline{g}$. The main result includes sharp upper and lower bounds for the quotient $d_f(z_0)/d_h(z_0)$, especially, for a family of locally univalent $Q$-quasiconformal harmonic mappings $f=h+\overline…
In this paper we extend Rado-Choquet-Kneser theorem for the mappings with weak homeomorphic Lipschitz boundary data and Dini's smooth boundary but without restriction on the convexity of image domain, provided that the Jacobian satisfies a certain boundary condition. The proof is based on a recent extension of Rado-Choquet-Kneser theorem by Alessandrini and Nesi \cite{ale} and it is used the approximation principle.
Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation $Δ(Δf)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $Δf=φ$ ($φ\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this paper is to prove that $f$ is Lipschitz continuos, and, further, it is bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|φ\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|φ\|_…
In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations ω. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated m…
This paper treats the class of normalized logharmonic mappings f(z) = zh(z)bar{g(z)} in the unit disk satisfying ϕ(z) = zh(z)g(z) is analytically typically real. Every such mapping f is shown to be a product of two particular logharmonic mappings, each of which admits an integral representation. Also obtained is the radius of starlikeness and an upper estimate for arclength. Additionally, it is shown that f maps the unit disk into a domain symmetric with respect to the real axis when it is univalent and its second dilatation has real coefficients.
We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general idea, for example, to construct log-harmonic Koebe mapping, log-harmonic right half-plane mapping and log-harmonic two-slits mapping and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings are obtained. Growth and distortion theorems for certain special subclass of log-harmonic mappings are…
We completely describe the asymptotic behaviour of the Riemann mapping function and its derivatives at an analytic cusp. We achieve the same for the inverse of the mapping function.
We study the problem of holomorphic extension of a smooth CR mapping from a real analytic hypersurface to a real algebraic set in complex spaces of different dimensions.
In this note, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete-Szegő problem for normalized concave functions with a prescribed pole in the unit disk.
In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(α)$ and $\mathcal{G}_H^0(β)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(α)$ and $F\in\mathcal{G}_H^0(β)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions…
In this paper, we study non sense-preserving harmonic mappings $f=h+\overline{g}$ in $\mathbb{D}$ when its analytic part $h$ is convex and injective in $\mathbb{D}$ and obtain radius of injectivety.
In this short note we apply certain iteration of the Janowski functions to estimate the integral means of some analytic and univalent mappings of $|z|<1$. Our method of proof follows an earlier one due to Leung \cite{YJ}.
We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in $\cc^2$ to a compact subset of $\cc^N$ which doesn't contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions.
The main purpose of this paper is to study the concept of normal function in the context of harmonic mappings from the unit disk $\mathbb{D}$ to the complex plane. In particular, we obtain necessary conditions for that a function $f$ to be normal.
In this paper we give a generalization and improvement of the Pavlović result on the characterization of continuously differentiable functions in the Bloch space on the unit ball in $\mathbb{R}^m$. Then we derive a Holland--Walsh type theorem for analytic normal mappings on the unit disk.
We continue the study of analytic functions in the unit disk of finite order with arbitrary set of singular points on the unit circle, introduced in \cite{FG}. The main focus here is made upon the inverse problem: the existence of a function from this class with a given singular set and zero set subject to certain Blaschke-type condition. We also discuss the local analog of the main result from \cite{FG} similar to the standard local Blaschke condition for analytic and bounded functions in the unit disk.
We establish a new characterization for a conformal mapping of the unit disk $\mathbb{D}$ to be convex, and identify the mappings onto a half-plane or a parallel strip as extremals. We also show that, with these exceptions, the level sets of $λ$ of the Poincaré metric $λ|dw|$ of a convex domain are strictly convex.
The subject considered in this paper has, at least, three points of interest. Suppose that we have a sequence of one-dimensional analytic varieties in a domain in $\Bbb C^n$. The cluster of this sequence consists from all points in the domains such that every neighbourhood of such points intersects with infinitely many different varieties. The first question is: what analytic properties does the cluster inherit from varieties? We give a sufficient criterion when the cluster contains an analytic disk, but it follows from examples of Stolzenberg and Wermer that, in general, clusters can contain …
The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization - a novel type of renormalization for holomorphic maps with a Siegel disk w…
We study the analytic continuation problem for a germ of a biholomorphic mapping from a non-minimal real hypersurface $M\subset\CC{n}$ into a real hyperquadric $\mathcal Q\subset\CP{n}$ and prove that under certain non-degeneracy conditions any such germ extends locally biholomorphically along any path lying in the complement $U\setminus X$ of the complex hypersurface $X$ contained in $M$ for an appropriate neighborhood $U\supset X$. Using the monodromy representation for the multiple-valued mapping obtained by the analytic continuation we establish a connection between nonminimal real hypersu…
We investigate harmonic mappings $f=h+\bar{g}$ defined in the unit disk, where $g$ and $h$ satisfy certain prescribed conditions and the analytic part $h$ belongs to the Ma and Minda class of starlike functions. Certain sharp radius results for univalency, close-to-convexity, fully starlikeness, fully convexity, and radius of strongly starlikeness are established. We also calculate the radius of uniformly starlikeness and convexity for these functions. Several results enhance the well-known radius results.
We obtain new characterizations of Bergman and Bloch spaces on the unit disc involving equivalent (quasi)-norms of these spaces.Our results are in spirit of estimates obtained by Fefferman and Stein for HArdy spaces in R^n.
The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and quasiconformal extensions. The extension is obtained by a reflection across the boundary of the surface using a family of Euclidean circles orthogonal to the surface. This gives a geometric generalization of the Ahlfors-Weill formula and extends the minimal surface. Thus one obtains a homeomorphism of $\overline{\mathbb{C}}$ onto a toplological sphere in $\overline{\mathbb{R}^3} = \mathbb{R}^3 \cup \{\infty\}$ that is real-…
Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.
For regular homeomorphisms of Sobolev class $W^{1,1}_{\textrm{loc}}$ having the Luzin $N$-property, it is established the estimation of the area of a disk image in terms of an angular dilatation. As a corollary, the analog of the well-known Ikoma-Schwartz lemma for such mappings is obtained.
We present a new proof of Pinchuk's theorem on the analytic continuation of a biholomorphic mapping from a strongly pseudoconvex analytic real hypersurface to a compact strongly pseudoconvex analytic real hypersurface in a complex euclidean space.
For $α> -1$ and $β>0, $ let $\mathcal{B}_{\mathcal{H}}^0(α, β)$ denote the class of sense preserving harmonic mappings $f=h+\overline{g}$ in the open unit disk $\mathbb{D}$ satisfying $|zh''(z)+α(h'(z)-1)|\leq β-|zg''(z)+αg'(z)|.$ First, we establish that each function belonging to this class is close-to-convex in the open unit disk if $β\in (0, 1+α]$. Next, we obtain coefficient bounds, growth estimates and convolution properties. We end the paper with applications and will construct harmonic univalent polynomials belonging to this class.
In this paper we study an algebraic and topological structure inside the following sets of special functions: Bloch functions defined on the open unit disk that are unbounded and analytic functions of bounded type defined a Banach algebra E into E, which are not Lorch-analytic.
A quantitative version of an inequality obtained in \cite[Theorem~2.1]{mathz} is given. More precisely, for normalized $K$ quasiconformal harmonic mappings of the unit disk onto a Jordan domain $Ω\in C^{1,μ} $ ($0<μ\le 1$) we give an explicit Lipschitz constant depending on the structure of $Ω$ and on $K$. In addition we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with $C^{2,α}$ boundary in terms of boundary function using the Hilbert transformations. Moreover it is given a sharp explicit quasiconformal constant in terms of the boundary fu…
We present a geometric approach to a well-known sharp inequality, due to Cowen and Pommerenke, about angular derivatives of general univalent self-maps of the unit disk.
We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of $\C^N$ and a real-algebraic CR-submanifold of $\C^{N'}$.
For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of harmonic mappings are shown to have finite Schwarzian norm.
Let $Ω\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $Ω$ and obtain a compactness result for Hankel operators on the Bergman space of $Ω$.
It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.
We consider CR submersive mappings between generic submanifolds in complex space. We show that, under suitable conditions on the manifolds, there is an integer k such that any jet of the CR mapping at a given point is a rational function of its k-jet at that point. As a consequence, it is shown that the stability group of a (suitably nondegenerate) real-analytic generic submanifold is a real algebraic Lie group. Moreover, it is shown that a formal equivalence between two such real-analytic submanifolds is necessarily convergent.
Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Examples are provided. This study is related to the problem of conformal welding and to representation of …
In this paper we prove that for an arbitrary pair $\{T_1,T_0\}$ of contractions on Hilbert space with trace class difference, there exists a function $\boldsymbolξ$ in $L^1({\Bbb T})$ (called a spectral shift function for the pair $\{T_1,T_0\}$ ) such that the trace formula $\operatorname{trace}(f(T_1)-f(T_0))=\int_{\Bbb T} f'(ζ)\boldsymbolξ(ζ)\,dζ$) holds for an arbitrary operator Lipschitz function $f$ analytic in the unit disk.
