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Clunie-Sheil-Small Conjecture

The Clunie-Sheil-Small conjecture proposes coefficient bounds for normalized, sense-preserving harmonic functions that are univalent functions on the unit disk. Write f(z)=h(z)+g(z)^_, where h(z)=z+sum_(n>=2)a_nz^n and g(z)=sum_(n>=2)b_nz^n are analytic functions. Sense preservation requires the Jacobian |h^'(z)|^2-|g^'(z)|^2 to be positive. The conjectured inequalities, for n>=2, are |a_n|<=(n+1)(2n+1)/6, |b_n|<=(n-1)(2n-1)/6, and ||a_n|-|b_n||<=n (Das 2026).

Das (2026) reported a counterexample to the coefficient-difference inequality: for each fixed K>1, a harmonic K-quasiconformal map has |a_n|-|b_n|=n^(1+epsilon_K) as n->infty for some epsilon_K>0. The same map satisfies both individual coefficient bounds, so this result does not refute those two assertions.

See also

Harmonic Function, Quasiconformal Map, Univalent Function

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References

Das, S. "A Counterexample to the Clunie-Sheil-Small Coefficient-Difference Conjecture via Quasiconformal Maps." 6 Oct 2026. https://arxiv.org/abs/2610.07770.

Cite this as:

Weisstein, Eric W. "Clunie-Sheil-Small Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Clunie-Sheil-SmallConjecture.html

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