Partition‐Based Refinements of the Hermite–Hadamard Inequality for Co‐Ordinated Convex Functions: Subdivision Weights, Cubature Bounds, and Higher‐Dimensional Extension

Source abstract

Existing two‐dimensional Hermite–Hadamard refinements typically rely on a single center‐point evaluation, and no partition‐based scheme exploiting the full symmetry of edge‐midpoints and quarter‐points has been systematized. In this paper, we develop a subdivision‐based refinement of the Hermite–Hadamard inequality for co‐ordinated convex functions on rectangular domains. By partitioning a rectangle into four equal subrectangles and applying the classical Hermite–Hadamard inequality on each subrectangle, we obtain a new upper bound that involves the values at the corners, the midpoints of the edges, and the center with weights 1/16, 1/8, and 1/4. We derive a matching lower bound involving the quarter‐points of the rectangle, and we prove that these weights are uniquely determined within the partition method (we do not claim global optimality among all inequalities using the same sampling points). We characterize the comparison with the Latif–Alomari bound: the two bounds coincide for additively separable convex functions, while explicit examples show that neither dominates universally. We then extend the framework to arbitrary dimension d , proving that the 2 d ‐subdivision yields a refined upper bound whose weights are governed by the face lattice of the hypercube. As illustrations, we apply the bounds to arithmetic, logarithmic, identric, and Stolarsky means in two variables, obtaining immediate corollaries of the main theorem for these special integrands; moreover, by applying the partition bounds to the integrands −ln( x y ) and 1/( x y ), we derive genuinely new two‐sided inequalities for the identric and logarithmic means, including an upper bound on L ( a , b ) that is provably sharper than the classical bound L ≤ A . We construct new composite quadrature rules for double integrals with a rigorous O ( n −2 + m −2 ) error estimate derived by repeated application of one‐dimensional error theory in each coordinate separately. Numerical experiments on both elementary integrands and integrands without an elementary closed‐form antiderivative confirm the theoretical convergence rate and demonstrate that the rigorous enclosures remain valid even when derivatives are unavailable. The results demonstrate that equal subdivision provides a systematic framework for deriving refinements of Hermite–Hadamard inequalities, constructing guaranteed cubature bounds, and extending the associated sampling structure to higher‐dimensional rectangular domains.