Dorff一个开放问题的反例
A counterexample to an open problem of Dorff
- Hunan First Normal University（湖南第一师范学院）
- Shenzhen Polytechnic University（深圳职业技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文构造一个保向调和微分同胚将单位圆盘映到椭圆，其自卷积在内部某点雅可比为零，从而否定回答了Dorff关于凸调和映射自卷积保持类别的开放问题。
AI中文摘要：
经典的Pólya-Schoenberg猜想由Ruscheweyh-Sheil-Small证明，断言两个归一化凸单叶函数的卷积仍然是凸的。这一性质不能推广到平面调和映射。2001年，Dorff提出了一个开放问题：具有有界像的归一化凸调和映射的自卷积是否必须保持在同一类中。我们构造了一个归一化的保向调和微分同胚，将单位圆盘映射到椭圆上；其自卷积在单位圆盘内部某点处雅可比行列式为零，这为Dorff的开放问题提供了否定答案。
英文摘要：
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.
