The main objective of this paper is a study of some new generalizations of Hilbert’s and Hardy–Hilbert’s type inequalities. We apply our general results to homogeneous functions. Also, we obtain the best possible constants when the parameters satisfy appropriate conditions.

By the method of weight coe cients and techniques of real analysis, a Hardy-Hilbert-type inequality with a general homogeneous kernel and a best possible constant factor is given. The equivalent forms, the operator expressions with the norm, the reverses and some particular examples are also considered.

In this paper, by the use of the weight coe cients, the transfer formula and the technique of real analysis, an extended multidimensional Hardy{Hilbert{type inequality with a general homogeneous kernel and a best possible constant factor is given. Moreover, the equivalent forms, the operator expressions and a few examples are considered.

Length-biased data are widely seen in applications. They are mostly applicable in epidemiological studies or survival analysis in medical researches. Here we aim to propose a Berry-Esseen type bound for the kernel density estimator of this kind of data. The rate of normal convergence in the proposed Berry-Esseen type theorem is shown to be O(n^(-1/6) ) modulo logarithmic term as n tends to infinity by a proper choice of the bandwidth. The results of a simulation study is also presented in this paper inorder to examine the performance of the result.