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We construct examples to show the sharpness of uniform continuity of quasiconformal mappings onto $s$-John domains. Our examples also give a negative answer to a prediction in [7].
It is well-known that several problems related to analysis on $s$-John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of $p$-capacity of a compact set $E$ and the central Whitney…
An effective algorithm is presented for solving the Beltrami equation df/dz = mu (df/dzbar) in a planar disk. The disk is triangulated in a simple way and f is approximated by piecewise linear mappings; the images of the vertices of the…
We introduce a weaker variant of the concept of three point property, which is equivalent to a non-linear local connectivity condition introduced in [12], sufficient to guarantee the extendability of a conformal map f from the unit disk…
It is shown that harmonic functions from a simply connected domain in R^3 to R^3 cannot always be expressed as a sum of a monogenic (hyperholomorphic) function and an antimonogenic function, in contrast to the situation for complex numbers…
Suppose that $f: \bR^n\to\bR^n$ is a mapping of $K$-bounded $p$-mean distortion for some $p>n-1$. We prove the equivalence of the following properties of $f$: doubling condition for $J(x,f)$ over big balls centered at origin, boundedness of…
We study conformal mappings from the unit disk to circular-arc quadrilaterals with four right angles. The problem is reduced to a Sturm-Liouville boundary value problem on a real interval, with a nonlinear boundary condition, in which the…
We introduce and study the $\rho$-Loewner energy, a variant of the Loewner energy with a force point on the boundary of the domain. We prove a large deviation principle for SLE$_\kappa(\rho)$, as $\kappa \to 0+$ and $\rho>-2$ is fixed, with…
We show that a domain that satisfies the visibility property with $\mathcal C^2$-smooth boundary is pseudoconvex.
We show that on convex domains with sufficiently smooth boundary the limit set of non-visible Kobayashi geodesics are contained in a complex face. In two dimensions, this implies the existence of a complex tangential line segment of…
It is proved for a strongly pseudoconvex domain $D$ in $\Bbb C^d$ with $\mathcal C^{2,\alpha}$-smooth boundary that any complex geodesic through every two close points of $D$ sufficiently close to $\partial D$ and whose difference is…
The configuration space of $k \geq 3$ ordered points in the Riemann sphere $\widehat{\mathbb C}$ is the Torelli space ${\mathcal U}_{0,k}$; a complex manifold of dimension $k-3$. If $m,n \geq 4$ and $F:{\mathcal U}_{0,m} \to {\mathcal…
A Lee-Yang polynomial $ p(z_{1},\ldots,z_{n}) $ is a polynomial that has no zeros in the polydisc $ \mathbb{D}^{n} $ and its inverse $ (\mathbb{C}\setminus\overline{\mathbb{D}})^{n} $. We show that any real-rooted exponential polynomial of…
It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of Joel Feinstein. Our example is R(K) for a certain compact…
We derive new integral estimates of the derivatives of mean $n$-valent functions in the unit disk. Our results develop and complement estimates obtained by E.P. Dolzhenko and A.A. Pekarskii, as well as recent inequalities obtained by the…
We give a new proof of the orientation reversibility of chordal Loewner energy by reversing the orientation of a chord in partial increments. This fact was first proved by Yilin Wang (arXiv:1601.05297) using the reversibility of chordal…
Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4\phi_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the…
A standard technique for producing monogenic functions is to apply the adjoint quaternionic Fueter operator to harmonic functions. We will show that this technique does not give a complete system in L2 of a solid torus, where toroidal…
A function $f$ from a domain in $\mathbb{R}^3$ to the quaternions is said to be inframonogenic if $\overline{\partial}\, f\overline{\partial} =0$, where $\overline{\partial} = \partial/\partial x_0+ (\partial/\partial…
We construct a set of quaternionic metamonogenic functions (that is, in $\mbox{Ker}(D+\lambda)$ for diverse $\lambda$) in the unit disk, such that every metamonogenic function is approximable in the quaternionic Hilbert module $L^2$ of the…