曾志红, 杨必成. 关于一个参量化的全平面Hilbert积分不等式[J]. 华南师范大学学报(自然科学版), 2017, 49(5): 100-103.
引用本文: 曾志红, 杨必成. 关于一个参量化的全平面Hilbert积分不等式[J]. 华南师范大学学报(自然科学版), 2017, 49(5): 100-103.
A Hilbert-Type Integral Inequality in the Whole Plane Related to the Exponential Function[J]. Journal of South China Normal University (Natural Science Edition), 2017, 49(5): 100-103.
Citation: A Hilbert-Type Integral Inequality in the Whole Plane Related to the Exponential Function[J]. Journal of South China Normal University (Natural Science Edition), 2017, 49(5): 100-103.

关于一个参量化的全平面Hilbert积分不等式

A Hilbert-Type Integral Inequality in the Whole Plane Related to the Exponential Function

  • 摘要: 探讨了一个新的参量化的全平面Hilbert积分不等式:建立2个权函数,利用积分的变量代换方法算得2个权函数为常数;运用带权的H lder不等式,通过恰当的配方,应用实分析技巧,引入非零的独立参量 以及满足关系 的参数 ,得到一个新的全平面非齐次核的具有最佳常数因子 且含中间变量的Hilbert型积分不等式,及其等价式和特殊参数下的齐次与非齐次核不等式.结果表明:这个新的参量化的全平面Hilbert积分不等式更优,是以往结果的推广;此文对Hilbert积分不等式的研究从第一象限拓展到全平面,为Hilbert不等式的纵深研究提供了更为广阔的思路,能丰富和完善Hilbert不等式的研究体系.

     

    Abstract: A new parametric Hilbert’s integral inequality in the whole plane is discussed. Two weight functions are established, and the two weight functions are calculated as constants by using the integral variable substitution method. By using the weighted H lder’s inequality, the appropriate formula, the real analysis techniques, and by introducing the non-zero independent parameters and the parameters satisfying , a new Hilbert-type integral inequality with intermediate variable, the best constant factor and non-homogeneous kernel are obtained, as well as its equivalent inequality and the inequalities with homogeneous or non-homogeneous kernel under special parameters. The result shows that this new parametric Hilbert’s integral inequality in the whole plane is superior, and is the expansion of the past results. The study of Hilbert’s integral inequality extends from the first quadrant to the whole plane in this paper, which provides a broader idea for the depth study of Hilbert’s inequality, enriches and perfects the study of Hilbert’s inequality system.

     

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