论文标题
哈迪的不平等及其后代
Hardy's Inequality and Its Descendants
论文作者
论文摘要
我们从随机变量方面提出并证明了Hardy不平等的概括（Hardy，1925），并表明它包含了Hardy不平等的通常（或熟悉）连续和离散形式的。接下来，我们改善了Hardy不平等的Li and Mao，具有一般的Borel措施和混合规范的权重，以意味着LIAO的离散版本和Hardy不平等，具有Muckenhoupt的权重以及由于Hardy和Hardy和Littlewood，Bliss和Bliss和Bradley而引起的混合规范版本。也给出了随机变量的等效公式。我们还通过随机变量制定了Hardy不平等，密切相关的Copson不平等，反向Copson不平等和Carleman-Pólya-Knopp不平等的反向版本。最后，我们将Copson的不平等与计数过程中的不平等联系起来和生存分析，并简要讨论其他应用。
We formulate and prove a generalization of Hardy's inequality (Hardy,1925) in terms of random variables and show that it contains the usual (or familiar) continuous and discrete forms of Hardy's inequality. Next we improve the recent version by Li and Mao of Hardy's inequality with weights for general Borel measures and mixed norms so that it implies the discrete version of Liao and the Hardy inequality with weights of Muckenhoupt as well as the mixed norm versions due to Hardy and Littlewood, Bliss, and Bradley. An equivalent formulation in terms of random variables is given as well. We also formulate a reverse version of Hardy's inequality, the closely related Copson inequality, a reverse Copson inequality and a Carleman-Pólya-Knopp inequality via random variables. Finally we connect our Copson inequality with counting process martingales and survival analysis, and briefly discuss other applications.
