PreprintCC BY 4.0arXiv
A counterexample to an open problem of Dorff
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.
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BibTeX
@misc{wang2026counterexample,
title = {{A counterexample to an open problem of Dorff}},
author = {Zhi-Gang Wang and Deguang Zhong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11556},
eprint = {2609.11556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11556},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A counterexample to an open problem of Dorff
AU - Zhi-Gang Wang
AU - Deguang Zhong
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11556
UR - https://arxiv.org/abs/2609.11556
AB - 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.
ER -