# Sharp Bounds on Arimoto's Conditional R\'{e}nyi Entropies Between Two Distinct Orders

## TL;DR

This paper derives precise bounds on Arimoto's conditional R'enyi entropy for different orders, identifying distributions that achieve these bounds and linking them to various information measures.

## Contribution

It provides the first sharp bounds on Arimoto's conditional R'enyi entropy for distinct orders and characterizes the distributions that attain these bounds.

## Key findings

- Identified distributions achieving the bounds.
- Derived bounds applicable to other information measures.
- Connected bounds to error probability and mutual information.

## Abstract

This study examines sharp bounds on Arimoto's conditional R\'enyi entropy of order $\beta$ with a fixed another one of distinct order $\alpha \neq \beta$. Arimoto inspired the relation between the R\'enyi entropy and the $\ell_{r}$-norm of probability distributions, and he introduced a conditional version of the R\'enyi entropy. From this perspective, we analyze the $\ell_{r}$-norms of particular distributions. As results, we identify specific probability distributions whose achieve our sharp bounds on the conditional R\'enyi entropy. The sharp bounds derived in this study can be applicable to other information measures, e.g., the minimum average probability of error, the Bhattacharyya parameter, Gallager's reliability function $E_{0}$, and Sibson's $\alpha$-mutual information, whose are strictly monotone functions of the conditional R\'enyi entropy.