Hardy's inequality and its descendants: a probability approach

We formulate and prove a generalization of Hardy's inequality [27] 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 [42] of Hardy's inequality with weights for general Borel measures and mixed norms so that it implies the discrete version of Liao [43] and the Hardy inequality with weights of Muckenhoupt [48] as well as the mixed norm versions due to Hardy and Littlewood [29], Bliss [8], and Bradley [14]. 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-Polya-Knopp inequality via random variables. Finally we connect our Copson inequality with counting process martingales and survival analysis, and briefly discuss other applications.