Sharp upper estimates and extremal maps

Determine whether the remaining argument for the bounded universal integral means spectrum can yield a sharp upper estimate or provide a description of the extremal conformal map.

Background

The paper establishes upper bounds for the bounded universal spectrum, including a strict inequality in the interior of the relevant interval. However, the authors state that the proof mechanism remains somewhat mysterious and explicitly leave unresolved whether it can attain a sharp upper bound or identify the conformal maps that realize extremal behavior.

References

It clearly can give better estimates, but it is not clear whether it can ever give a sharp estimate from above or provide description of the extremal map.

— Brennan's Conjecture and the Universal Integral Means Spectrum  (2610.23853 - Binder et al., 16 Oct 2026) in Section 1.3, p. 2