Papers

Suppose \(a, b, c\) are polynomials with \(ab \neq 0\) and \(c\) nonconstant. We determine all entire solutions to the equation \begin{equation*} f'f” - a(z)f^2 = b(z)e^{2c(z)}. \end{equation*} This completely solves two open questions posed by Gundersen and Yang in [Comput. Methods Funct. Theory 21 (2021), 605-617].

We give a geometric characterization of the pluricomplex Poisson kernel of bounded strongly linearly convex domains in $\mathbb C^{n+1}$ in terms of their foliation by all complex geodesic discs whose closure contains a fixed boundary point, thus confirming a conjecture posed by Bracci et al. in 2009, even in greater generality. The proof relies crucially on our previous results obtained in a series of two papers. We also revisit the homogeneous complex Monge–Ampère equation associated with the pluricomplex Poisson kernel and show that its solutions are far from unique in general by constructing a family of pairwise nonproportional continuous solutions on the unit ball in $\mathbb C^{n+1}$. This sharply contrasts with the uniqueness of solutions to the corresponding equation for the pluricomplex Green function.

We develop a discrete $q$-Hermitian Clifford calculus based on coordinatewise Jackson differences on multiplicative $q$-lattices. The resulting Hermitian Jackson–Dirac operators are nilpotent, factor a $q$-Laplacian, and have occupancy-dependent Euler anticommutators. These give explicit Fischer projectors and homotopies, but the local calculus is not controlled by total degree. We prove a two-face Cauchy–Kovalevskaya theorem for polynomials and extend it to a $q$-analytic Jackson–Fischer class. A divided-power conjugation recovers scalar Euler relations and transfers the classical joint Fischer decomposition and dimension formulas. The Jackson–Fischer completion is a vector-valued $q$-Fock space whose polarized nullspaces have projected reproducing kernels; the joint kernel is obtained by alternating projections. We also determine the linear symmetry group of the multiplicative lattice.

The Gromov hyperbolicity constant of a metric space \((X,d)\) is the infimum of all \(\delta\ge0\) such that \((X,d)\) is \(\delta\)-hyperbolic. For a Kobayashi hyperbolic domain \(\Omega\subset\C^n\) and a boundary point \(p\in\partial\Omega\), let \(\delta_{\mathrm{loc}}(\Omega,p)\) denote the local Gromov hyperbolicity constant obtained by restricting the points to arbitrarily small Euclidean neighborhoods of \(p\), while distances are still measured by the ambient Kobayashi distance \(K_\Omega\). For each even integer \(M\ge2\), let \(\mathfrak H(M)\) be the supremum of these constants over all complex dimensions \(n\ge2\) and all domains whose boundary is smooth and convex near the distinguished point and has D'Angelo type at most \(M\) at that point. We prove \[ \frac{\log 2}{2} M \le \mathfrak H(M) < 36M. \] Thus the optimal universal dependence of the local Gromov hyperbolicity constant on boundary type is linear. We also show that no analogous bound holds for the global Gromov hyperbolicity constant, even among smooth bounded strongly convex domains.

In this paper, we prove an asymptotic expansion of the $L^2$-extension index of a smooth Hermitian metric on a holomorphic vector bundle, in which the Chern curvature appears as the second-order coefficient. By using this expansion, we show in a unified way that there is an equivalence between how sharp the $L^2$-extension is and how positive or negative the curvature is. We also introduce a new notion of $q$-$L^2$-extension indices and investigate partial positivity and flatness in terms of these indices.

In this paper, we show that partial positivity is not preserved by marginalization in both the real and complex settings. We also give a sufficient condition for the preservation of partial positivity under marginalization.

The Komlós conjecture is a classic problem in discrepancy theory; it asks whether an absolute constant $K$ exists such that given any $n$ vectors $a_1,\ldots,a_n$ inside the $m$-dimensional Euclidean ball, regardless of how large $m,n$ are, there is always a selection of signs $\varepsilon_1,\ldots,\varepsilon_n$ guaranteeing $$\|\varepsilon_1a_1+\ldots+\varepsilon_na_n\|_\infty \leq K.$$ We show that if the $\varepsilon_i$'s are allowed to take not just the values of $\pm 1$ but any unit modulus complex number, which we refer to as complex discrepancy, then the above inequality holds for a finite, explicit constant $K_{\mathbb{C}}$. Here, the $\ell^\infty$ norm of the resulting vector in $\mathbb{C}^m$ is the largest modulus of its entries, and thus the complex discrepancy of real vectors is equivalent to their rank-$2$ vector discrepancy. This quantity provides an upper bound (modulo uniform constant prefactor) on Gaussian discrepancy – a discrepancy measure introduced by Chewi, Gerber, Rigollet and Turner. Thus, we also resolve the Komlós conjecture for Gaussian discrepancy. Our paper builds upon the recent work of Bansal and Jiang on the Beck-Fiala and Komlós conjectures, which we approach from the formalism of Burkholder and the Bellman function method from probability and harmonic analysis. Our work was in part motivated by the realization that the complex discrepancy of the columns of any unitary matrix is equal to 1, a fact that follows from a straightforward calculation based on Idel and Wolf's generalization of the Sinkhorn normal form for unitary matrices.

Let \(\eta=(\eta_n)_{n\ge0}\) be a complex sequence such that \(F_\eta(z)=\sum_{n\ge0}\eta_nz^n\in\Hol(\D)\), and let \(\mathcal R_{(\eta)}\) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of \(\mathcal R_{(\eta)}:\mathcal D^p_\alpha\to\mathcal D^q_\beta\) for \(1<p,q<\infty\) and \(\alpha,\beta>-1\). The characterization is formulated in terms of the \(H^q\)-norms of the dyadic blocks of \(F_\eta\). For \(-1<\alpha<p-2\), boundedness and compactness coincide and reduce to \(F_\eta\in\mathcal D^q_\beta\). For \(\alpha=p-2\) and for \(\alpha>p-2\), boundedness is characterized by membership in analytic truncated Besov spaces and analytic Besov spaces, respectively; compactness has the same characterization for \(q<p\) and is characterized by the corresponding little spaces for \(p\le q\). We also obtain operator norm and essential norm estimates. As applications, we obtain boundedness and compactness characterizations for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures. In particular, we characterize \(C_\mu:B^p\to B^p\), answering a question left open by Sun, Ye and Zhou and later noted by Tang. For each \(p>2\), we also construct a symbol \(F_\eta\in H^\infty\cap\lambda^p_{1/p}\) for which \(\mathcal R_{(\eta)}\) is not bounded on \(H^p\).

We initiate a joining rigidity theory for rational maps on the Riemann sphere $\mathbb P^1=\mathbb P^1(\mathbb C)$. Let $f_1,f_2\colon\mathbb P^1\to\mathbb P^1$ be rational maps of degree at least $2$, and $\mu_1,\mu_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,\mu_1)$ and $(J(f_2),f_2,\mu_2)$, namely ergodic probability measures on $J(f_1)\times J(f_2)$ which are invariant under $f_1\times f_2$ and whose marginals are $\mu_1$ and $\mu_2$. Our main theorem shows that a positive-mass local holomorphic relation forces algebraic rigidity. More precisely, if the joining charges the graph of a local biholomorphism, then that local relation globalizes to an invariant algebraic curve and yields either a finite cycle of rational graph or transpose-graph relations, or a genuinely multi-valued invariant algebraic correspondence. If no local biholomorphic graph has positive joining measure, then the joining generates a compact non-discrete family of local holomorphic relations. The proof introduces normalized inverse branch transfer maps and studies their cluster limits. Starting from a local biholomorphic graph of positive joining measure, recurrence and contraction of inverse branches produce recurrent local intertwining relations. These are promoted to an algebraic relation by a local-to-global rigidity argument in the non-Lattès case and by affine uniformization in the Lattès case. In the absence of any positive-mass local biholomorphic graph, the cluster family must be infinite, and its non-discrete closure gives the second alternative.

We give geometric characterizations of analytically linearizable nondegenerate Siegel foliations in $\C^2$ using complex tangencies with shrinking strictly pseudoconvex hypersurfaces. Our principal result treats a fixed bounded smoothly bounded strictly pseudoconvex Reinhardt domain: in coordinates compatible with the eigendirections of the linear part, two-dimensional tangency loci along a shrinking sequence characterize analytic Siegel linearizability. For round spheres we prove a stronger statement: no a priori alignment of the Euclidean coordinate axes with the eigendirections is required; the tangency hypothesis itself forces the linear part to be unitarily diagonalizable with negative real eigenvalue ratio. We also establish a polynomial Shilov-boundary criterion on a round sphere and a logarithmic one-boundary criterion valid for arbitrary smooth strictly pseudoconvex domains, provided the embedded closed bidisc lies on the pseudoconvex side. Finally, we study the transverse-holomorphic dynamics of smooth tangency tori, including a rigidity theorem in the periodic case.

We construct two families of compact complex threefolds fibred over $\mathbb P^1$ by two-dimensional complex tori. For each positive integer $r$ and each parameter $c$ in a fixed half-plane, the families $Y_{r,c}$ and $Z_{r,c}$ have two multiple fibres of multiplicities $(4,6)$ and $(3,6)$, respectively, and one reduced normal-crossings fibre whose normalization consists of $r$ del Pezzo surfaces of degree six. Their fundamental groups are $\mathbb Z/2$ and $\mathbb Z/3$, respectively, and both have Euler characteristic $2r$. We compute their integral homology; the index-one members are rational homology six-spheres. Every member has algebraic dimension one, lies outside Fujiki's class $\mathcal C$, and has connected automorphism group $\mathbb C^*$. For fixed type and index, two members are biholomorphic exactly when their parameters differ by an element of $(6/r)\mathbb Z$. Each underlying smooth six-manifold therefore supports uncountably many pairwise non-biholomorphic complex structures. The arithmetic input classifies finite monodromy pairs and their integral lifts to a parabolic Jacobi group acting on a rank-four lattice. A quadratic form controls integral lifting and, on the rank-two square-zero locus, computes the monodromy saturation index. We prove that the relevant invariant tensors persist under finite-index restriction, which extends the construction to arbitrary compact base curves. We also construct translation twists after base change, establish a local periodic toroidal quotient theorem in arbitrary dimension, and determine the additive Bockstein spectral sequences of the original threefolds.

We study singularities of rational inner functions (RIFs) $\phi$ in three and more variables. We focus on curve and higher-dimensional singularities of $\phi$ in $\mathbb{T}^n$ and we show that, under certain conditions, derivative integrability conditions of partial derivatives of $\phi=\frac{q}{p}$ stay constant along irreducible components of the zero set of $p$, and that the presence of vertical line singularities does not affect global integrability. This answers Question 2 in arXiv:1906.10913. Those results are then generalized to $n$ variables. Finally, we consider slice matrices, which we use to obtain derivative integrability conditions of unidirectional compositions $\phi^N$. We present several new examples, including an RIF with two vertical line singularities and one with vertical surface singularities.

Let $X$ be a compact connected Riemann surface equipped with two nonconstant meromorphic projections $q,z:X\to\mathbb{P}^1$. Their differentials determine a canonical meromorphic relative differential morphism $Z:T_q=q^*T\mathbb{P}^1\to T_z=z^*T\mathbb{P}^1$, whose local coefficient in coordinate frames is $Z=z_q\,\partial_z\otimes dq$. Its divisor is $\operatorname{Div}(Z)=R_z-R_q$, and its local logarithmic derivative defines a rank-one logarithmic connection on $L=\operatorname{Hom}(T_q,T_z)$. The general theory of logarithmic connections is classical. The main point of the present paper is that, in the dual conformal-net setting, the Levi–Civita curvature data of the two nets realize this logarithmic connection by the symmetric formula $H^{-1}\widetilde{\omega}\,dz-H\omega\,dq =2i\,d\log z_q =2i\,Z^{-1}dZ$, where the last expression is understood through local trivializations of $L$. Thus the two dual conformal connection differentials occur with opposite signs, and their difference is the logarithmic Maurer–Cartan form of the relative differential morphism. Its residues recover the relative ramification multiplicities, $\operatorname{Res}_p\Theta=2i(r_z(p)-r_q(p))$, and globally $\sum_{p\in X}\operatorname{Res}_p\Theta =4i(\operatorname{deg}z-\operatorname{deg}q)$. Thus local conformal connection data determine both relative ramification and the difference of the degrees of the two projections. We also discuss the dual morphism, the square-root issue, and a hyperelliptic example.

We classify the generators of stability-preserving semigroups on polynomial spaces with bounded coordinate degrees. The generators have differential order at most two, with principal coefficients characterized by low-degree nonnegativity conditions. In disk coordinates, positive degree damping gives strict zero-freeness and a sharp spectral gap, extending the Hermitian gap of Bravyi, Gosset, Liu, and Wong to complex generators. The resulting analytic domain yields deterministic classical algorithms for ground energies and stable product-state queries for bounded-degree Suzuki-Fisher Hamiltonians with bounded local strength and fixed positive fields. Field perturbation gives zero-field energy-value approximation schemes for bounded-degree unweighted EPR and bipartite Quantum MaxCut. We also prove that, up to scalars, the Hermitian qubit Hamiltonians whose full Gibbs tensors are Lee-Yang at every temperature in a fixed basis are precisely the edgewise phase-rotated Suzuki-Fisher family. With strictly positive longitudinal fields, these tensors are zero-free on a polydisk of radius greater than one at each positive inverse temperature. These two statements answer questions of Wong, Bravyi, Gosset, and Liu. Degree-preserving, coefficient-positive stability semigroups have concave sector growth rates, yielding token-graph concavity. We also give a counterexample to a proposed explicit ground-state radius.

In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.

In this paper, we study some geometric properties of the disk bundles over the complete Kähler manifolds. When the base space is the complete Kähler-Einstein manifold, we compute the holomorphic sectional curvature, Riemannian sectional curvature of the complete Kähler-Einstein metric on the disk bundle. Moreover, we present a parametric criteria for whether the manifold admits negative pinched property. When the base space is a bounded pseudoconvex domain equipped with its complete Kähler metric, the corresponding ball bundle in the $k$-direct sum of line bundles is a bounded pseudoconvex Hartogs domain with fiber dimension $k$ . If the base domain is simply-connected and $k\geq2$, we prove that the Bergman metric on such a Hartogs domain is Kähler-Einstein if and only if the domain is biholomorphically equivalent to a unit ball.

We establish quantitative comparisons between the Laplace–Beltrami spectra of Kähler metrics of equal area on a compact Riemann surface. An estimate in terms of the oscillation of a potential gives bounds for fractional powers of reciprocal eigenvalues and explicit intervals for eigenvalue ratios. Bounds involving the gradient and Laplacian of the potential give further comparisons. On the Riemann sphere, we obtain estimates for individual eigenvalues, reciprocal sums and counting functions. For a Kähler metric on the unit 2-sphere which is centered in the sense that the unit normal vector field integrates to zero, the Dirichlet energy of the potential yields quantitative improvements of Hersch's bounds for the first positive eigenvalue and the sum of the first three reciprocal eigenvalues.

Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\mathcal{F}$ and $\mathcal{G}$ near $C$, with $C$ not $\mathcal{G}$-invariant, we attach to every point $p \in C$ an index $\mathrm{Ind}(\mathcal{F},\mathcal{G},C,p) \in \mathbb{C}$ of $\mathcal{F}$ relative to the reference $\mathcal{G}$, and we prove that these indices add up to $T\mathcal{G} \cdot C$. When $C$ is $\mathcal{F}$-invariant the index is the Camacho–Sad index corrected by the order of tangency of $\mathcal{G}$ with $C$, and the index formula reduces to the Camacho–Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\mathcal{G}$ with $C$, let $k(\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \sum_{p} \frac{1}{1 - k(\mathcal{G},C,p)} = \frac{2}{3}\left( C \cdot C + g - 1 \right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\mathcal{F}$ and $\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\mathcal{F}$ with $\mathcal{G}$ along $C$ equals the number of tangencies of $\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\mathcal{F}$.

In this paper, a Hard Lefschetz theorem for logarithmic sheaves twisted by pseudo-effective line bundles is established, extending a theorem of Demailly, Peternell, and Schneider.

This paper deals with a certain family of series which is derived from generating functions of the Frobenius-Euler and Apostol type polynomials and numbers. When special cases of this series are examined, it is shown that it generates some well-known families of zeta functions, such as the Hurwitz zeta and the Lerch zeta, etc. We also give an open problem for these relation. We give analytic continuation of this function. By using the Cauchy Residue Theorem, We show that this series interpolates the Frobenius-Euler type polynomials at negative integers. Moreover, by applying the Laplace transform to the generating function for these polynomials, we get infinite series representations involving the Lerch zeta function and these polynomials. In addition, We show that this interpolation function can be representation in terms of the Lerch zeta function and the Hurwitz zeta function. Some special values for the results are also given.

We introduce a generalized Cauchy–Kowalevski extension associated with a biaxial decomposition of $\mathbb{R}^{m+1}$, allowing monogenic functions to be reconstructed from data prescribed on the subspace $\{X\in\mathbb{R}^{m+1}: z=0\}$ where $z=x_0+x_1e_1$. The extension admits a natural decomposition into monogenic layers, leading to operators $\mathrm{CK}_{z,k}$. Classes of biaxial monogenic functions generated by these operators are characterized by systems of partial differential equations, and explicit power series solutions are obtained. The action of the operators $\mathrm{CK}_{z,k}$ on homogeneous polynomials is then investigated, leading to a class of biaxial monogenic plane waves and the associated system of partial differential equations.

We introduce a projective-kernel framework for the study of replicable functions in modular function theory. The main point is that the same two-point kernel simultaneously encodes the differential projective geometry of a modular function and the Faber–Grunsky data governing its replicability. This makes it possible to translate between Schwarzian invariants, coefficient identities, and modular correspondences within a single structure. The kernel yields a reconstruction theorem showing that the ordinary Schwarzian determines the normalized Grunsky matrix and hence the underlying Laurent expansion. When Norton's replicability relations are imposed, the projective kernel produces strong arithmetic restrictions on the possible cusp data. In the degree-one case this leads to a precise classification: replicability and complete replicability are equivalent to solvability of the projective monodromy, while the icosahedral case is excluded by an explicit Grunsky obstruction. The same principle extends to the arithmetic Hecke triangle groups. The projective-kernel viewpoint also isolates the remaining global difficulty in Norton's Hauptmodul conjecture and provides a natural setting in which replicability, projective monodromy, and modular differential invariants can be studied together.

A formal solution to a partial differential equation of infinite order involving the action of Mahler-type operators in several variables is obtained. Such formal solutions is constructed by means of a fixed point argument applied to the integral representation of the deceleration operator in several variables that has been developed. The Laplace transform in two variables of the actual solution of a related problem in both Borel plane and Fourier space turns out to admit strong asymptotic expansion along adequate multidirections.

In this paper, we introduce some weighted reduced quaternionic modules induced by a reduced quaternionic Cauchy-Riemann operator associated to arbitrary orthogonal bases of R3. A weighted conformal invariant property of the modules is given.

Let $X$ be a hyperbolic Riemann surface and let $\mu$ and $\nu$ be laminations on $X$ homotopic to measured foliations with finite Dirichlet integral. We prove that $\mu$ and $\nu$ are filling if and only if there exists a homeomorphism to another Riemann surface $f:X\to Y$ and an integrable holomorphic quadratic differential $q$ on $Y$, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of $q$ respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual $\mathbb{R}$-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two $\mathbb{R}$-trees. We construct examples such that $f:X\to Y$ is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.

Let $\Omega\subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and let $\psi\in SH_m(\Omega)$ be a fixed negative $m$-subharmonic function. In this paper we introduce a relative finite energy class $\mathcal E_{m,\psi}(\Omega),$ which may be viewed as a Hessian analogue of the relative energy classes appearing in the pluripotential theory of complex Monge–Ampère equations. We develop a systematic pluripotential theory in this setting. More precisely, we introduce a relative Hessian capacity associated with the prescribed singularity type $\psi$, construct relative mixed Hessian products, and establish their fundamental properties. We prove a monotone convergence theorem and a Bedford–Taylor type continuity theorem for Hessian measures in the class $\mathcal E_{m,\psi}(\Omega)$. A central result of the paper is a relative comparison principle, which yields uniqueness of solutions to complex Hessian equations with prescribed singularities. As an application, we establish an existence and uniqueness theorem for the equation $(dd^c u)^m\wedge\beta^{n-m} =\mu $ for a large class of positive Radon measures that do not charge $m$-polar sets. The results obtained here provide a relative finite energy framework for complex Hessian equations and extend several fundamental aspects of Cegrell's theory to the setting of prescribed singularity types.

Let $\pi:Z\rightarrow Y$ be a smooth projective morphism of complex manifolds with connected rationally connected fibers. We prove holomorphic approximation on arbitrary compact sets and finite-jet interpolation on arbitrary closed discrete sets for continuous liftings defined on open Riemann surfaces and holomorphic near those sets. For smooth projective morphisms of smooth complex algebraic varieties and algebraic base maps from smooth affine curves, the approximating liftings can be chosen algebraic, with interpolation on any finite set. In both cases the resulting lifting is homotopic to the initial one through continuous liftings of the fixed base map. For connected smooth projective complex manifolds, this gives the equivalence between the algebraic Oka-1 property and rational connectedness. Every rationally connected smooth projective complex manifold is also Oka-1.

We prove that every primitive harmonic $(n-1,1)$-form of a closed Kähler manifold of complex dimension $n$ vanishes provided its Kähler curvature operator is $\frac{n^2-n+2}{2}$-positive if $n \geq 4$, respectively $\frac{7}{2}$-positive if $n=3$. As a consequence, a closed Kähler manifold of complex dimension $n \geq 3$ with $3$-positive Kähler curvature operator is a real cohomology complex projective space.

Inspired by constructions of entire curves in Oka geometry, we construct rational curves on complex Fano manifolds analytically, by alternately deforming a holomorphic disk to reduce its area and enlarging its source. Positive Ricci curvature yields an area-decreasing deformation, while an affine-lift perturbation enlarges the source at a controlled area cost. The two operations change the area and the weighted derivative by amounts we estimate explicitly, and analytic compactness passes from the resulting disks to a holomorphic sphere. The main result produces a sphere that meets a prescribed compact fiber without being contained in it, with an explicit area bound.

The classical Pólya-Schoenberg conjecture, proved by Ruscheweyh-Sheil-Small, asserts that the convolution of two normalized convex univalent functions is again convex. This property fails to carry over to planar harmonic mappings. In 2001, Dorff posed the open problem whether the self-convolution of a normalized convex harmonic mapping with bounded image must remain in the same class. We construct a normalized sense-preserving harmonic diffeomorphism that maps the unit disk onto an ellipse; its self-convolution has vanishing Jacobian at some interior point of the unit disk, which provides a negative answer to Dorff's open problem.