Brennan's conjecture states that the derivative of a conformal mapping 
from a simply
connected proper plane domain onto the unit
disk satisfies
 |
for 
,
where 
is area measure. The range of exponents is sharp. Equivalently, the universal
integral means spectrum satisfies 
(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 
for 
for the bounded universal integral
means spectrum. Their announced strict improvement for 
with 
has its details deferred to a separate paper. Binder et
al. (2026) state that some authors used AI to help understand the original argument,
but that their own write-up was prepared without AI assistance.
