Probability Theoretic Generalizations of Hardy's and Copson's Inequality

A short proof of the classic Hardy inequality is presented for $p$-norms with $p>1$. Along the lines of this proof a sharpened version is proved of a recent generalization of Hardy's inequality in the terminology of probability theory. A probability theoretic version of Copson's inequality is discussed as well. Also for $0<p<1$ probability theoretic generalizations of the Hardy and the Copson inequality are proved.