离散型Hilbert不等式的推广及应用

On an Extension of the Discrete Type Hilbert Inequality and Its Application

通过引入两个参数,构造了一个离散的分式型的核函数,并由此建立相应的Hilbert不等式。利用余切函数的部分分式展开,证明了所构建的不等式的常数因子可用余切函数表示,且常数因子是最佳的。通过对参数赋值,得到了一些有趣的特殊结果。

By introducing two parameters, a discrete kernel function of fraction type is constructed, and the corresponding Hilbert inequality is established. By using the partial fraction expansion of the cotangent function, it is proved that the constant factor of the constructed inequality can be expressed by cotangent function, and the constant factor is optimal. Finally, by specifying the values of parameters, some interesting special results are obtained.