Fully resolve the Boros–Moll ratio log-concavity conjecture

Determine an explicit threshold M for which the sequence of Boros–Moll ratio terms u_i(m)=d_{i-1}(m)d_{i+1}(m)/d_i(m)^2 is strictly log-concave for all m≥M, and verify the remaining finite values of m so as to establish the log-concavity conjecture for every m≥2.

Background

The paper proves that the ratio sequence {u_i(m)} is strictly log-concave for all sufficiently large m, but the proof does not provide an explicit value of the threshold M. Consequently, the original conjecture asserting log-concavity for every m≥2 remains unresolved for the finite range below the eventual-asymptotic threshold. The authors indicate that tracking the constants in the interior and edge analyses could yield an effective threshold, after which the remaining finite cases could be checked directly.

References

The proof of Theorem \ref{eventual_LC} is stated asymptotically and does not provide an explicit value of $M$. The estimates in our analysis arise from explicit rational functions and quantitative contraction arguments, which might suggest that an effective version could be obtained by tracking the constants throughout the interior, left-edge and right-edge regimes. Such an effective threshold, combined with a verification of the remaining finite cases, would fully resolve Conjecture \ref{LC_BM_conj}.

— On Two Conjectures Related to the Boros-Moll Sequences  (2609.10234 - Aravinda, 9 Sep 2026) in Section 5, Final Remarks