A Novel Multidimensional Refinement of the Half-Discrete Hardy–Hilbert Inequality with a Parameterized Kernel and a Partial Sum Term

This paper introduces a novel multidimensional half-discrete Hardy–Hilbert-type inequality that simultaneously addresses several key extensions in the literature. The inequality incorporates a general parameterized kernel involving a scalar term and the β-norm of a vector, and replaces the traditional discrete coefficient with a partial sum. Under suitable parameter conditions, the resulting inequality is sharper and preserves the optimal constant factor. The proof employs a systematic combination of weight-function techniques, parameter introduction, real-analysis methods, and the Euler–Maclaurin summation formula. Equivalent characterizations of the best possible constant are provided, and several meaningful corollaries are deduced, thereby unifying and generalizing a series of earlier inequalities.