多参数复合核的Hilbert型不等式及应用

On a Hilbert-type Inequality Involving a Composite Kernel with Several Parameters and Its Applications

引入若干参数,构造一个与指数函数和对数函数相关、并且同时包含齐次和非齐次两种情形的复合型核函数.借助分析的有关技巧,建立了一个新的Hilbert型积分不等式,并证明了常数因子是最佳的.作为定理的应用,对参数赋予特殊数值,最后给出了一些特殊结果.

In this work,by introducing some parameters,a composite kernel function which is related to exponential function and logarithmic function,and contains both homogeneous and non-homogeneous cases is constructed.By means of some techniques of real analysis,a new Hilbert type integral inequality is established,and it is proved that the constant factor is optimal.In particular,as an application of the theorem,special values are given to the parameters,and some special results are established at the end of the paper.