# On some conjectures by Lu and Wenzel

## TL;DR

This paper investigates several conjectures related to inequalities in matrix theory, proving some in special cases, providing bounds, and offering new proofs for existing inequalities.

## Contribution

It proves Conjecture 2 and related conjectures in certain cases, introduces new equivalent conjectures, and improves bounds for Conjecture 3.

## Key findings

- Proved Conjecture 2 in some cases
- Established a larger upper bound for Conjecture 3
- Provided new simple proofs of the complex BW inequality

## Abstract

In order to give a unified generalization of the BW inequality and the DDVV inequality, Lu and Wenzel proposed three Conjectures 1, 2, 3 and an open Question 1 in 2016. In this paper we discuss further these conjectures and put forward several new conjectures which will be shown equivalent to Conjecture 2. In particular, we prove Conjecture 2 and hence all conjectures in some special cases. For Conjecture 3, we obtain a bigger upper bound $2+\sqrt{10}/2$, and we also give a weaker answer for the more general Question 1. In addition, we obtain some new simple proofs of the complex BW inequality and the condition for equality.