Hardy's Inequality and Its Descendants

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-P\'olya-Knopp inequality via random variables. Finally we connect our Copson inequality with counting process martingales and survival analysis, and briefly discuss other applications.