This study explores a new multidimensional Hilbert-type inequality involving one partial sum, by utilizing transfer formula and Hermite-Hadamard's inequality. The kernel 1/(u(m)+parallel to v(k)parallel to(alpha))(lambda) (lambda > 0) in the new inequality has two general internal variables compared 1 with previous work, and the best value is achieved with certain parameters. Finally, the equivalent forms, the operator expressions and some particular cases are presented.

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 1/ ( ln m+‖ v(k)‖ _α ) ^β( β >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.

By using the techniques of real analysis, a new more accurate Mulholland-type inequality with two different internal variables involving two partial sums is given. The equivalent conditions of the best value related to several parameters are provided, and some particular inequalities are deduced.

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 beta-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.

By means of the weight functions, the idea of introduced parameters, and the reverse Hardy’s integral inequality, an extended reverse Hardy-type inequality is obtained, and then a new reverse half-discrete Hilbert-type inequality with the general homogeneous kernel, as well as multiple lower limit function and remainder sum, is given. The equivalent statements of the best value related to several parameters are considered. As applications, the equivalent forms and some particular examples are provided.

By means of the weight functions, the idea of introduced parameters and Hardy’s integral inequality, a multidimensional half-discrete Hilbert-type inequality with a general homogeneous kernel involving one derivative function of m-order is obtained. The equivalent statements of the best value in the new inequality related to several parameters are considered, and some corollaries are deduced.

In this paper, by employing the Euler–Maclaurin summation formula and real analysis techniques, an improved version of the parameterized Hardy–Hilbert inequality involving two partial sums is established. Based on the obtained inequality, the equivalent conditions of the best possible constant factor related to several parameters are discussed. Our results extend the classical Hardy–Hilbert inequality and improve certain existing results.

In this paper, by using the techniques of real analysis, with the help of the Euler–Maclaurin summation formula, Abel’s summation by parts formula, and the differentiation mid-value theorem, we establish a half-discrete Hardy–Mulholland-type inequality involving one multiple upper limit function and one partial sum. Based on the obtained inequality, we characterize the condition of the best possible constant factor related to several parameters. At the end of the paper, we illustrate that some new half-discrete Hardy–Mulholland-type inequalities can be deduced from the special values of the parameters. Our results enrich the current results in the study of half-discrete Hardy–Mulholland-type inequalities.

By means of the weight functions, the idea of introduced parameters and using the techniques of real analysis, a multidimensional Hardy-Hilbert's integral inequality involving one derivative function of higher-order is obtained. As applications, the equivalent statements of the best possible constant factor in the new inequality related to several parameters are considered. Some corollaries are obtained.

In this paper, by introducing a general homogeneous kernel function and several parameters, we establish a new Hardy–Hilbert-type integral inequality involving two derivative functions of higher-order. For the resulting inequality, we determine the equivalent conditions of the best possible constant factor related to the parameters. As applications, we demonstrate that a lot of new Hardy–Hilbert-type integral inequalities can be derived by choosing specific homogeneous kernel functions.

In this book, by applying the weight functions, the idea of introduced parameters and the techniques of real analysis and functional analysis, we use some lemmas and then provide a new Hilbert-type integral inequality with the nonhomogeneous kernel and the best possible constant factor. As applications, some new Hardy-Hilbert’s integral inequalities with two interval variables involving extended derivative functions of higher-order and extended multiple upper limit functions are obtained. The equivalent statements of the best possible constant factors related to several parameters are given.

By using the weight coefficients and the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given. The equivalent statements of the best value related to several parameters are provided. The equivalent forms, some inequalities in particular parameters, and the operator expressions are obtained.

This paper presents a new half-discrete multidimensional Hilbert-type inequality involving one higher-order derivative function utilizing transfer formula and Hermite-Hadamard's inequality. The inequality investigates a general intermediate variable in kernel 1/ (x +Ilv (k)Il(alpha)() lambda+ m) (x,lambda > 0) than previous work. The research explores the best value related to certain parameters. Finally, the equivalence forms and operator expressions are also presented.