Development of general kernels of half-discrete Mulholland-type inequalities and Hilbert-type inequalities with extensions and reverse form

Half-discrete Hilbert-type inequalities form an important class of analytic inequalities with broad applications in numerical analysis, approximation theory, and optimization. However, most existing studies focus on inequalities with homogeneous kernels, leaving a significant research gap in the development of inequalities involving nonhomogeneous or general kernels. Moreover, extensions that include higher-order derivative function or multiple upper limit function remain insufficiently explored. Therefore, this study addresses these gaps by formulating and analyzing new half-discrete Mulholland-type and Hilbert-type inequalities with general kernels. The objectives are to construct novel inequalities with general nonhomogeneous kernels, determine their best values, and derive their operator expressions, equivalent forms, and reverse inequalities. Furthermore, the study extends these inequalities to higher-order derivative and multiple upper limit functions. The research employs methods from real analysis and inequality theory, combined with weight function to construct and analyze new forms of half-discrete inequalities. These extend the mathematical framework of half-discrete inequalities and their reverse forms by developing new general kernels. In conclusion, this study successfully develops new inequalities involving general nonhomogeneous kernels and their reverse forms while accommodating higher-order derivatives and multiple upper limit functions. These findings enrich the half-discrete inequality theory with potential applications in mathematical analysis, functional analysis, and applied fields such as physics and signal processing