For every normalized univalent harmonic mapping f=h+overlineg of the unit disk D, with expansions h(z)=z+∑_n=2^∞a_nz^n and g(z)=∑_n=2^∞b_nz^n, one has || a_n|-| b_n||≤ n for every integer n≥2.

Clunie–Sheil-Small coefficient-difference conjecture

Clunie and Sheil-Small proposed the conjecture in 1984 for normalized univalent harmonic mappings, including the bound $\left|\lvert a_n\rvert-\lvert b_n\rvert\right|\le n$ . It remained open for the full class.

Known results

The conjecture was verified for several geometric subclasses, including starlike, close-to-convex, convex, typically real, and convex-in-one-direction mappings (Ponnusamy and Kaliraj, 2015).
Ponnusamy and Kaliraj proved related coefficient bounds for a larger subclass, but not the full class.
Later work proved the conjectured bounds for subclasses with constant second dilatation and for several $K$ -quasiconformal subclasses; the full class remained open.

October 2026 claimed counterexample

Suman Das's new preprint claims that, for every $K>1$ , a quasiconformal construction has coefficient differences of order $n^{1+\varepsilon K}$ , violating the conjectured bound while retaining individual coefficient estimates. The claim is presently unverified.

Current status (as of October 2026): The conjecture is claimed to be disproved by a new preprint, but absent verification the full problem remains unsettled.

Clunie–Sheil-Small coefficient-difference conjecture

Primary source

Additional references

Progress summary

Solutions 0