经典离散型Hilbert型不等式的推广及应用  

Generalization and Applications of a Classical Discrete Hilbert-Type Inequality

作　　者：有名辉[1] YOU Minghui(Mathematics Teaching and Research Section,Zhejiang Institute of Mechanical and Electrical Engineering,Hangzhou 310053,Zhejiang,China)

机构地区：[1]浙江机电职业技术学院数学教研室,浙江杭州310053

出　　处：《汕头大学学报（自然科学版）》2020年第4期27-34,共8页Journal of Shantou University：Natural Science Edition

基　　金：浙江省教育厅一般科研资助项目(Y201737260);浙江机电职业技术学院科教融合一般项目(A-0271-20-007)。

摘　　要：构造了一个新的含有参变量的离散型齐次核函数,并通过将核函数分解为两个单调递减函数的乘积证明其是单调的,从而实现对权函数的估计并建立相应的Hilbert齐次型不等式.类似的方法还建立了非齐次核的Hilbert型不等式.根据余切函数的有理分式展开,给出了所建立不等式的常数因子的三角函数表示,并且证明了常数因子是最佳的.最后,赋予参数一些特殊的数值,得到了一些经典的结果和其它一些有意义的特殊不等式.In this paper,a new discrete homogeneous kernel function with parameter variables is constructed,and it is proved to be monotonic by decomposing the kernel function into two functions.In this way,the weight function can be estimated and the Hilbert-type inequality with homogeneous kernel is established.Similarly,the Hilbert-type inequality with non-homogeneous kernel can also be obtained.According to the rational fraction expansion of the cotangent function,the trigonometric function expression of the constant factor in the obtained inequality is given,and the constant factor is proved to be the best possible.Finally,by giving the parameters special values,some classical results and other meaningful special results are obtained.

关 键 词：HILBERT不等式 有理分式展开 余切函数 离散型 

分 类 号：O174.1[理学—数学]