含混合核的Hilbert型不等式及应用

A Hilbert-Type Inequality Involving a Mixed Kernel and Its Application

通过引入多个参数,构建一个与对数函数和指数函数有关、并且同时包含齐次和非齐次两种形态的混合核函数.由此建立与新构建的核函数相关联的Hilbert型积分不等式,且常数因子是最佳的.通过对参数赋予特殊的数值,证明了新建立的不等式是已有一些结论的推广,并给出若干新的含有Catalan常数等特殊常数的Hilbert型不等式.

By introducing several parameters,a mixed kernel function related to exponent and logarithmic functions and including both homogeneous and non-homogeneous forms was constructed.Then,the Hilbert type integral inequality associated with the new kernel was established,and the constant factor was proved to be the best possible.By assigning special values to the parameters,the newly obtained inequality proves to be an extension of some existing results.Furthermore,some new special Hilbert-type inequalities with the constant factors related to special constants such as Catalan constant are presented at the end of the paper.