Using weight functions, we establish a few equivalent statements of two kinds of Hardy-type integral inequalities with nonhomogeneous kernel in the whole plane. The constant factors related to the extended Hurwitz-zeta function are proved to be the best possible. In the form of applications, we deduce some special cases involving homogeneous kernel. We additionally consider some particular inequalities and operator expressions.

中文翻译：

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与扩展的Hurwitz-zeta函数有关的整个平面上的Hardy型积分不等式

使用权重函数，我们建立了两个Hardy型积分不等式在整个平面上具有非齐次核的等价陈述。与扩展的Hurwitz-zeta函数有关的常数被证明是最好的。以应用程序的形式，我们推断出一些涉及同构内核的特殊情况。我们还考虑了一些特殊的不等式和运算符表达式。