Sharp Comparisons of Jensen Gaps under Atom-Mass Constraints

We prove sharp comparisons of Jensen gaps under lower atom-mass constraints. For a convex function (f), write (J_f(X)=\mathbb E f(X)-f(\mathbb E X)). The classical Jensen-variance problem compares (J_f(X)) with (\operatorname{Var}(X)=J_{x^2}(X)). We extend this framework by comparing two Jensen gaps (J_f(X)) and (J_g(X)). Under coordinatewise monotonicity of the relative second divided difference (R_{f,g}(x,y,z)=f[x,y,z]/g[x,y,z]), we obtain sharp two-sided constants for finite atomic random variables whose atom masses are bounded below by (\alpha). The sharp constants are given by two-point laws with masses (\alpha) and (1-\alpha). The variance case is recovered by taking (g(x)=x^2). Applications include power gaps, exponential gaps, power divergences, Hellinger distance, KL versus chi-square comparisons, shifted powers, reciprocal powers, and positive mixtures.