Hermite–Hadamard Inequalities for Subharmonic Functions on $\mathbb {R}^{n}$: A Potential-Theoretic Approach

In this study, we establish a generalized framework for Hermite–Hadamard-type inequalities for subharmonic functions on $$\mathbb {R}^{n}, \ n\ge 2.$$ R n , n ≥ 2 . Moving beyond traditional convex analysis, we utilize potential-theoretic methods (specifically, Dirichlet solutions and Poisson kernels), to derive integral inequalities over annular and rectangular domains. We provide a unified measure-theoretic interpretation that clarifies and extend recent results, demonstrating that the classical Hermite–Hadamard inequality is a specific one-dimensional manifestation of broader potential-theoretic principles. By employing approximation techniques, we relax standard regularity requirements, extending the applicability of these results to general subharmonic functions.