The Krzyż conjecture asserts that every analytic function 
from the unit disk to the punctured unit
disk satisfies 
for every positive integer 
. The proposed extremal functions are
 |
where 
.
Wikström (2026) proved the conjecture, including the equality characterization, for 
. The proof combines an analytic reduction with rigorous
computer-assisted exclusion of extremals with fewer than six atoms in their integral
representation. The general conjecture remains open.
Wikström (2026) reports using Claude for numerical exploration and verification-code development, with a separate GPT-6 Astra code review. The author checked the arguments and code.
Xie (2026) proved the seventh-coefficient case, including the same equality characterization with 
. The proof combines an
analytic reduction with finite rational inequalities excluding extremizers supported
on at most six atoms. Wikström (2026, p. 2) independently proved the same
case by validated elimination of masses in the integral representation followed by
computer-assisted exclusions.
MacDonald (2026, pp. 2-3) proved Peretz's conjecture that any 
maximizing 
in the class of bounded zero-free analytic
functions above has 
, for every positive
integer 
.
