一个多参数的Hilbert型不等式及应用

A Multi-parameter Hilbert-type Inequality and Its Application

引入多个参数,构建一个同时包含齐次和非齐次,且由指数函数与对数函数相复合的积分核函数.通过对齐次和非齐次两种形态进行统一处理,建立一个新的Hilbert型积分不等式,并证明这一新的不等式的常数因子是最佳值.另外,文末还给出了赋予参数特殊数值所得到的若干推论,其中包括核函数为对数函数与双曲余切函数复合的Hilbert型不等式.

By introducing several parameters, a integral kernel function which contains both homogeneous and non-homogeneous equation, exponential function and logarithmic function is constructed.By unifying the two forms of homogeneous and non-homogeneous, a new Hilbert-type integral inequality is established, and it is proved that the constant factor of the new inequality is the best value.In addition, by giving special values to the parameters, some interesting special results are presented at the end of paper, including Hilbert-type inequality whose kernel function is a composite of logarithmic function and hyperbolic cotangent function.