# $L^p$-norm inequality using q-moment and its applications

## TL;DR

This paper establishes new inequalities relating Lp-norms in Euclidean space using q-moments of escort distributions, with applications to entropy bounds and probability estimates.

## Contribution

It introduces Lp-norm inequalities based on q-moments for Euclidean spaces, extending previous finite measure results and applying them to entropy and probability bounds.

## Key findings

- Derived upper bounds for Renyi and Tsallis entropies using q-moments.
- Established inequalities between two Renyi entropies.
- Provided bounds for probabilities of subsets in Euclidean space.

## Abstract

For a measurable function on a set which has a finite measure, an inequality holds between two Lp-norms. In this paper, we show similar inequalities for the Euclidean space and the Lebesgue measure by using a q-moment which is a moment of an escort distribution. As applications of these inequalities, we first derive upper bounds for the Renyi and the Tsallis entropies with given q-moment and derive an inequality between two Renyi entropies. Second, we derive an upper bound for the probability of a subset in the Euclidean space with given Lp-norm on the same set.