用权函数方法、参量化思想及实分析技巧,建立一个新的齐次核为1/(x+y)λ+1(λ>0)的具有一个导函数的Hardy-Hilbert型积分不等式,给出联系该不等式的最佳常数因子及多参数的等价性质,并给出非齐次核的类似情形及若干特例.

引入独立参数,应用实分析的技巧及权函数方法,建立一个一般非齐次核逆向的Hilbert型积分不等式.提出了它的等价形式及联系逆式的常数因子最佳性与多参数的等价陈述.作为应用,给出若干特殊核的例子.

利用实分析和权函数的方法,研究非齐次核Hilbert型积分不等式的若干等价条件,证明联系Hurwitz zeta函数的常数因子为最佳值.

应用权函数的方法及参量化思想,给出一个齐次核为反正切函数的较为精确的具有最佳常数因子的半离散Hilbert型不等式,同时给出了等价形式及特殊结果.

利用实分析和权函数的方法,研究第一类非齐次核Hardy型积分不等式的若干等价条件,证明联系Gamma函数的常数因子为最佳值.

By means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered.

Innovation is to lead the development of the first power.In the teaching of Higher Mathematics,introducing mathematics experiments,playing the role of basic experiment and exploration, the application and comprehensive modeling of the experimental, the reform of college mathematics courses, can cultivate innovative talents.Measures should be taken to develop the mathematical experiment in the teaching of Higher Mathematics:to maintain higher mathematics knowledge system,to complete the curriculum setting and management,to improve the teaching content,methods,survey analysis comparison table,to formulate scientific way of evaluation,to achieve knowledge innovation,to cultivate students′comprehensive quality,innovation ability and innovative talents.

By introducing two pairs of conjugate exponents and using the improved Euler-Maclaurin's summation formula for estimating the weight coefficient, we obtain a new basic Hilbert-type inequality and prove that constant factor is the best possible. As applications, we build its equivalent form and some particular results.

By introducing a parameter and estimating the weight coefficient, we obtain a new Hilbert-type integral inequality with a composite kernel and a best constant factor. As applications, we also consider its equivalent forms and reverse forms.