几个基本Hilbert型不等式的统一推广

On a Unified Extension of Several Basic Hilbert-type Inequalities

通过引进多个参数,构建一个第一象限内的非齐次核函数,并由此建立了具有最佳常数因子的Hilbert型不等式.通过简单的变量替换,把非齐次型的Hilbert型不等式演化为齐次型.通过对参数赋值,并借助余切函数的有理分式展开,得到最佳常数因子与余切函数高阶导数关联的Hilbert型不等式,推广了若干基本的Hilbert型不等式和一些经典结果.

By introducing several parameters,a non-homogeneous kernel function in the first quadrant was constructed,and the corresponding Hilbert-type inequality with the optimal constant factor established.By means of simple variable substitution,the non-homogeneous Hilbert-type inequality was transformed into a homogeneous type.By specifying parameters and using the rational fraction expansion of the cotangent function,the Hilbert-type inequalities with the optimal constant factor related to the higher derivative of the cotangent function were obtained,which generalizes some basic Hilbert type inequalities and some classical results.