On the Leibniz rule for random variables

TL;DR

This paper establishes a Leibniz-type inequality relating the spread of random variables to their $L_p$-norms, inspired by inequalities in analysis and operator theory.

Contribution

It introduces a new inequality connecting the spread of random variables with their $L_p$-norms, extending concepts from Kato-Ponce and Rieffel's strong Leibniz property.

Findings

Proves a Leibniz-type inequality for random variables.

Links the spread of variables to their $L_p$-norms.

Provides a theoretical foundation inspired by analysis inequalities.

Abstract

We prove a Leibniz-type inequality for the spread of random variables in terms of their $L_p$-norms. The result is motivated by the Kato-Ponce inequalities and Rieffel's strong Leibniz property.