Prescribed-prefix curvature

Determine whether every finite forest F satisfies \Delta_k(F)=i_k(F)^2-i_{k-1}(F)i_{k+1}(F)\ge0 at every internal index k<\lceil(2\alpha(F)-1)/3\rceil.

Background

The paper proves strict log-concavity through the larger prefix \lfloor17\alpha(F)/25\rfloor for sufficiently large forests, while the classical decreasing tail begins at approximately 2\alpha(F)/3. The stated problem asks whether the entire prefix before that tail is log-concave for every forest, without an order-size restriction.

This problem is stronger than barrier log-concavity because the prescribed prefix can include indices after the actual mode, where the coefficients are descending.