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Brennan's Conjecture

Brennan's conjecture states that the derivative of a conformal mapping g:Omega->D from a simply connected proper plane domain onto the unit disk satisfies

int_Omega|g^'(z)|^pdA(z)<infty

for 4/3<p<4, where dA is area measure. The range of exponents is sharp. Equivalently, the universal integral means spectrum satisfies B(-2)=1 (Binder et al. 2026).

Binder et al. (2026) validated an OpenAI proof and gave a substantially shorter argument proving the conjecture. They also obtained the bound B_b(t)<=t^2/4 for -2<=t<=2 for the bounded universal integral means spectrum. Their announced strict improvement for -2<t<2 with t!=0 has its details deferred to a separate paper.

See also

Conformal Mapping, Univalent Function, Universal Integral Means Spectrum

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References

Binder, I.; Ivrii, O.; Prause, I.; and Smirnov, S. "Brennan's Conjecture and the Universal Integral Means Spectrum." 16 Oct 2026. https://arxiv.org/abs/2610.23853.Brennan, J. E. "The Integrability of the Derivative in Conformal Mapping." J. London Math. Soc. 18, 261-272, 1978. https://doi.org/10.1112/jlms/s2-18.2.261.

Cite this as:

Weisstein, Eric W. "Brennan's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrennansConjecture.html

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