A new multidimensional Hilbert-type inequality involving one partial sum

We derive a novel multidimensional Hilbert-type inequality incorporating a partial sum, by employing transfer formulas and the Hermite–Hadamard inequality. Unlike existing results, this new inequality features a kernel of the form $\frac{1}{\left ( \ln m+\left \Vert v(k)\right \Vert _{\alpha }\right ) ^{\beta }}( \beta >0)$, where $v(k)$ is a generalized internal variable. Moreover, we prove the optimality of the inequality under specific parameter conditions. Additionally, we discuss the equivalent formulation of the inequality and its operator expression. These results extend the applicability of Hilbert-type inequalities in multidimensional analysis.