Related papers: Always-convex harmonic shears

A harmonic mapping $f=h+\overline{g}$ in $\mathbb{D}$ is $\varphi$-normal if $f^{\#}(z)=\mathcal{O}(|\varphi(z)|), \text{ as } |z|\to 1^-,$ where $f^{\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish several sufficient…

Given a bounded n-connected domain in the plane bounded by non-intersecting Jordan curves, and given one point on each boundary curve, L. Bieberbach proved that there exists a proper holomorphic mapping of the domain onto the unit disc that…

In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic…

Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings…

Using several complex variables techniques, we investigate the interplay between the geometry of the boundary and compactness of Hankel operators. Let f be a function smooth up to the boundary on a smooth bounded pseudoconvex domain D in…

We discuss a general result of holomorphic extension of a real analytic function $f$ defined on the boundary $\partial D$ of a real analytic strictly convex subset $D\subset\subset \C^n$. We show that this follows from the hypothesis of…

Dorff et al. [4], proved that the harmonic convolution of right half-plane mapping with dilatation -z and mapping f_\beta = h_\beta + \bar{g}_\beta, where f_\beta is obtained by shearing of analytic vertical strip mapping, with dilatation…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of…

We show that every strictly pseudoconvex domain $\Omega$ with smooth boundary in a complex manifold $\mathcal{M}$ admits a global defining function, i.e., a smooth plurisubharmonic function $\varphi \colon U \to \mathbb R$ defined on an…

Let $n \geq 3$ and $\Omega$ be a bounded domain in $\mathbb{C}^n$ with a smooth negative plurisubharmonic exhaustion function $\varphi$. As a generalization of Y. Tiba's result, we prove that any holomorphic function on a connected open…

Let $\mathbb{X}$ be a Jordan domain satisfying hyperbolic growth conditions. Assume that $\varphi$ is a homeomorphism from the boundary $\partial \mathbb{X}$ of $\mathbb{X}$ onto the unit circle. Denote by $h$ the harmonic diffeomorphic…

Let D be a piecewise smooth bounded convex Reinhardt domain in C^2. Assume that the symbols f and g are continuous on the closure of D and harmonic on the disks in the boundary of D. We show that if the product of Hankel operators H^*_f H_g…

We show that for every connected analytic subvariety $V$ there is a pseudoconvex set $\Omega$ such that every bounded matrix-valued holomorphic function on $V$ extends isometrically to $\Omega$. We prove that if $V$ is two analytic disks…

We show that each pseudoconvex domain $\Omega\subset {\mathbb C}^n$ admits a holomorphic map $F$ to ${\mathbb C}^m$ with $|F|\le C_1 e^{C_2 \hat{\delta}^{-6}}$, where $\hat{\delta}$ is the minimum of the boundary distance and…

We extend a classical result of Caughran/Schwartz and another recent result of Gunatillake by showing that if D is a bounded, convex domain in n-dimensional complex space, m is a holomorphic function on D and bounded away from zero toward…

Let $D$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$ and let $f$ be a positive continuous function on $\partial D$. Under some assumptions on $\varphi$, it is shown that the problem $\Delta u=2\varphi(u)$ in $D$ and $u=f$ on $\partial…

In this paper, we establish some Schwarz type lemmas for mappings $\Phi$ satisfying the inhomogeneous biharmonic Dirichlet problem $ \Delta (\Delta(\Phi)) = g$ in $\mathbb{D}$, $\Phi=f$ on $\mathbb{T}$ and $\partial_n \Phi=h$ on…

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…

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into…

Let $\Omega\subset\mathbb R^2$ be a chord arc domain. We give a simple proof of the the following fact, which is commonly known to be true: a nontrivial harmonic function which vanishes continuously on a relatively open set of the boundary…