# Wehrl-type coherent state entropy inequalities for $SU(1,1)$ and its $AX+B$ subgroup

## TL;DR

This paper investigates Wehrl-type entropy inequalities for the groups $SU(1,1)$ and $AX+B$, proving cases for the affine group and linking the general problem to an unresolved issue in complex analysis.

## Contribution

It proves Wehrl-type entropy inequalities for the $AX+B$ group at even integer $p$ and relates the broader conjecture to an open problem in analytic function theory.

## Key findings

- Proved Wehrl-type inequalities for $AX+B$ at even integer $p$
- Reduced the general $AX+B$ case to an open problem in complex analysis
- Connected entropy inequalities to properties of analytic functions on the upper half plane

## Abstract

We discuss the Wehrl-type entropy inequality conjecture for the group $SU(1,1)$ and for its subgroup $AX+B$ (or affine group), their representations on $L^2({\mathbb R}_+)$, and their coherent states. For $AX+B$ the Wehrl-type conjecture for $L^p$-norms of these coherent states (also known as the R\'enyi entropies) is proved in the case that $p$ is an even integer. We also show how the general $AX+B$ case reduces to an unsolved problem about analytic functions on the upper half plane and the unit disc.