We prove sharp variance bounds for Jensen gaps of finite atomic distributions under lower atom-mass constraints. The constants are obtained by reducing the extremal problem to two-point laws with one atom of minimal admissible mass. The results are compared with classical unrestricted mean-variance Jensen bounds and with Bregman-quotient bounds. Applications are given to Jensen-Nesbitt type inequalities, reciprocal power functions, harmonic and logarithmic mean refinements, power sums, finite-alphabet f-divergence comparisons, and ensemble ambiguity gaps for convex losses.
Denis Sheremet (Tue,) studied this question.