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Universal Integral Means Spectrum

The universal integral means spectrum is the supremum of the growth exponents of derivative integral means over univalent functions on the unit disk. For a univalent function f, define

M_t(f,r)=int_0^(2pi)|f^'(re^(itheta))|^tdtheta
(1)

and

beta_f(t)=limsup_(r->1^-)(lnM_t(f,r))/(ln(1/(1-r))).
(2)

The universal spectrum B(t)=sup_(f)beta_f(t) allows all univalent functions, whereas B_b(t) restricts to bounded univalent functions. They obey B(t)=max{B_b(t),3t-1} (Binder et al. 2026).

Binder et al. (2026) proved

B_b(t)<=(t^2)/4
(3)

for -2<=t<=2, implying Brennan's conjecture. They additionally announce strict inequality for -2<t<2 with t!=0, but defer the details of that strengthening to a separate paper. Binder et al. (2026) derived their shorter argument while validating earlier AI-generated proofs.

See also

Brennan's Conjecture, Conformal Mapping, Univalent Function

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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.

Cite this as:

Weisstein, Eric W. "Universal Integral Means Spectrum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniversalIntegralMeansSpectrum.html

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