Generalized extremal conjecture for admissible functionals

Establish that, for every integer n≥1 and every admissible functional F_p in the generalized extremal problem, the extremal value G_n(p) equals 2/e and the unique extremal function is f_n(z)=exp((z^n−1)/(z^n+1)).

Background

The paper introduces admissible functionals of the form F_p(f)=2p(a_0,a_n)a_0+(1−p(a_0,a_n))a_n on the restricted class of singular inner functions with at most n atoms and satisfying 2a_0≤a_n. These functionals interpolate between maximizing the nth coefficient, as in the Krzyż problem, and maximizing the constant term.

The authors prove that the generalized statement follows whenever the original Krzyż conjecture holds, and they verify it for n≤5. They formulate the all-n assertion as a conjecture, leaving its proof unresolved for the remaining cases.

References

Since the Krzy.z conjecture is believed to be true by many experts, we expect the following more general conjecture for problem eq:extr-alt-p to be true. Let n\ge 1 and let $\F_p$ be an admissible functional. Then $G_n(p)=2/e$ and the only extremal function for eq:extr-alt-p is $f_n(z)= e{\frac{zn-1}{zn+1}$.

— On extremal problems related to the Krzyż conjecture  (2610.23817 - Instanes et al., 16 Oct 2026) in Section 5.2, “Equivalent formulations of the generalized conjecture”