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

TL;DR

This paper presents new probability theoretic generalizations of Hardy's and Copson's inequalities, including sharpened and extended versions for different p-norm ranges, with simplified proofs and broader applicability.

Contribution

It introduces novel probability-based generalizations of Hardy's and Copson's inequalities, including for 0<p<1, with simplified proofs and improved bounds.

Findings

Sharpened version of Hardy's inequality for p>1

Generalizations for 0<p<1

Discussion of probability theoretic Copson's inequality

Abstract

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.