Work

A counterexample to an open problem of Dorff

Year: 2026

Type: preprint

Abstract: 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.11556

Authors: Wang, Zhi-Gang, Zhong, Deguang

Language: English

Topic: Analytic and geometric function theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Education Department of Hunan Province, Basic and Applied Basic Research Foundation of Guangdong Province

Awards: G4343562384, G553955554, G5632167411

OpenAlex ID: W7212244187

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