In this work, by introducing several parameters, a new kernel function including both the homogeneous and non-homogeneous cases is constructed, and a Hilbert-type inequality related to the newly constructed kernel function is established. By convention, the equivalent Hardy-type inequality is also considered. Furthermore, by introducing the partial fraction expansions of trigonometric functions, some special and interesting Hilbert-type inequalities with the constant factors represented by the higher derivatives of trigonometric functions, the Euler number and the Bernoulli number are presented at the end of the paper.

中文翻译：

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关于一些 Hilbert 型不等式的新推广

在这项工作中，通过引入几个参数，构造了一个包括齐次和非齐次情况的新核函数，并建立了与新构造的核函数相关的希尔伯特型不等式。按照惯例，还考虑了等效的 Hardy 型不等式。此外，通过介绍三角函数的部分分数展开，文末介绍了一些特殊而有趣的希尔伯特型不等式，其常数因子由三角函数的高阶导数、欧拉数和伯努利数表示。