Let D=z∈C:|z|<1. If f=h+overlineg:D→C is a normalized sense-preserving harmonic mapping with bounded convex image, where h(z)=z+∑_n≥2a_nz^n and g(z)=∑_n≥2b_nz^n, must its harmonic self-convolution ff, defined by (ff)(z)=z+∑_n≥2a_n^2z^n+overline∑_n≥2b_n^2z^n, also be a normalized convex harmonic mapping in the same class?

Dorff’s open problem on self-convolution of convex harmonic mappings

Dorff’s 2001 problem asks whether self-convolution preserves the class of convex harmonic mappings. The claimed refutation concerns the harmonic problem, not the classical Pólya–Schoenberg theorem.

Known results

2009: Certain harmonic convolutions are convex in one direction under local-univalence or dilatation hypotheses; local univalence cannot generally be omitted.
2013: Convolution criteria and examples were obtained, with no general conclusion that the result lies in $\mathcal{K}_{H}^{0}$ .
2018: A corrected and more general formulation of a Dorff problem was claimed solved for specified parameter families.
2020: A paper gave a partial answer to a Dorff problem using univalence and directional-convexity conditions.

September 2026 counterexample claim

On September 10, 2026, a report identified Zhi-Gang Wang and Deguang Zhong’s preprint A counterexample to an open problem of Dorff, which claims an explicit construction refuting preservation under self-convolution. The construction and its implication remain unverified in the retrieved material.

Current status (as of September 2026): The preservation conjecture is claimed false by Wang and Zhong’s preprint, but the counterexample has not been independently verified; earlier partial and corrected-version results remain established as reported.

Dorff’s open problem on self-convolution of convex harmonic mappings

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