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Deguang Zhong

Publications and source records attributed to Deguang Zhong.

8 recordsLinked to original sources
arXiv2026

A counterexample to an open problem of Dorff

The classical P\'olya-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.

math.CV↗
arXiv2026

Meromorphic solutions of first-order differential equations with rational exponential coefficients

We study first-order differential equations $f'=R(e^z,f)$, where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $R$ is a polynomial of degree at most two in its second variable. Such an algebraic solution necessarily has the form $S(e^{z/q})$, with $S$ rational and $q$ a positive integer, giving an affirmative answer to Question~6.2 in Gundersen's collection. The geometric input is Guillot's theorem on single-valued trajectories of meromorphic vector fields on surfaces. The additional argument compares the fibration provided by that theorem with the original exponential coordinate. A ramification calculation forces every finite nonzero branch value $a$ to satisfy $Da=a$, which excludes such a value and reduces the comparison to a two-point cover. Divisor divisibility and relative algebraic closedness then recover a Riccati equation over the original coefficient field. No growth hypothesis is imposed. We also identify the algebraic degree with the least integer deck period and describe the growth and value fibres of the resulting solutions.

math.CV↗
arXiv2026

Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps

We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,\pi]\to\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \[ (T_kF)(\xi)=\int_{\mathbb S^{n-1}} k\!\left(\arccos\langle\xi,\eta\rangle\right) F(\eta)\,d\sigma(\eta), \] where $\sigma$ is normalized surface measure. For every measurable $F:\mathbb S^{n-1}\to\overline{\mathbb B^n}$, we prove \[ w\!\left(\operatorname{co}T_kF(\mathbb S^{n-1})\right) \le2\lambda_1(k), \] where $w$ denotes mean width, normalized by $w(\overline{\mathbb B^n})=2$, and $\lambda_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(\eta)=Q\eta$ almost everywhere, with $Q\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\mathbb B^n)|\le\omega_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.

math.CV↗
arXiv2026

Global convergence and monotonicity of Newton iteration for the inverse Gr\"otzsch modulus

Let \[ \mu(r)=\frac{\pi}{2}\frac{\Kc(r')}{\Kc(r)}, \qquad r'=\sqrt{1-r^2},\qquad 0 \pi/2$, and whether it is strictly increasing when $y>\pi$. We prove both assertions. The main point is that $\mu$ has exactly one inflection point $a\in(1/2,1/\sqrt2)$. If the zero lies in the convex region, the Newton iterates increase to it from the left. If the zero lies in the concave region, the orbit crosses the inflection point, overshoots the zero at most once, and then decreases to the zero. For $y>\pi$ we obtain the stronger estimate $0<x_n<x_{n+1}<\mu^{-1}(y)<3-2\sqrt2<1.$

math.CV↗
arXiv2026

A Pogorelov-type counterexample to the discreteness and openness of gradient mappings

Let $\Omega\subset\mathbb{R}^{n}$ be a domain, and suppose that $u\in W^{2,n}_{loc}(\Omega)$ satisfies the following ineqiality \[ \det D^2u\geq \delta>0\qquad\text{a.e. in }\Omega. \] A question of Guerra--Tione \cite[Question 5.5]{GuerraTione} asks whether the gradient mapping $Du$ must be open and discrete. In this paper, we give an explicit Pogorelov-type construction showing that the answer is negative in every dimension $n\geq4$: there exists $u\in W^{2,n}_{loc}(\Omega)$ satisfying the above lower bound for the Hessian determinant, with $D^2u>0$ a.e., such that $Du$ collapses an entire line segment to a single point and hence is not discrete. We also show that the same construction has a logarithmic divergence when $n=3$ and therefore does not directly settle the three-dimensional case.

math.AP↗
arXiv2026

Quasiregular values from generalized manifold with controlled geometry

The main aim of this paper is to establish the Reshetnyak's theorem for quasiregualr values from generalized $n$-manifold with suitable controlled geometry to Euclidean space $\mathbb{R}^{n}.$ This generalizes a previous result due to Kangasniemi and Onninen on the setting of Euclidean space [A single-point Reshetnyak's theorem, Trans. Amer. Math. Soc., 378(2025): 3105-3128].

math.CV↗
arXiv2025

Optimal estimates for mappings admitting general Poisson representations in the unit ball

Suppose that $1<p\leq\infty$ and $φ\in L^{p}(\mathbb{B}^{n},\mathbb{R}^{n}).$ In this note, we use Hölder inequality and some basic properties of hypergeometric functions to establish the sharp constant $C_{p}$ and function $C_{p}(x)$ in the following inequalities $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}}\cdot||φ||_{L^{p}}$$ and $$|u(x)|\leq \frac{C_{p}(x)}{(1-|x|^{2})^{(n-1)/p}}\cdot||φ||_{L^{p}},$$ where $u$ are those mapping from the unit ball $\mathbb{B}^{n}$ into $\mathbb{R}^{n}$ admitting general Poisson representations. The obtained results generalize and extend some known results from harmonic mappings (\cite[Proposition 6.16]{ABR92} and \cite[Theorems 1.1 and 1.2]{DM12}) and hyperbolic harmonic mappings (\cite[Theorems 1.1 and 1.2]{CJLK20}).

math.CV↗