On norm inequalities related to the geometric mean

TL;DR

This paper establishes new inequalities involving the geometric mean of positive definite matrices and unitarily invariant norms, confirming a conjecture for the case of two matrices and broadening understanding of matrix norm inequalities.

Contribution

It proves a conjectured inequality for two positive definite matrices under unitarily invariant norms, extending previous results and confirming the conjecture in this specific case.

Findings

Proved inequality for m=2, p≥1, r≥1, for all unitarily invariant norms.

Confirmed the conjecture posed by Dinh, Ahsani, and Tam for the case m=2.

Connected the inequalities to recent results by Audenaert.

Abstract

Let $A_i$ and $B_i$ be positive definite matrices for all $i=1,\cdots,m.$ It is shown that $$\left|\left|\sum_{i=1}^m(A_i^2\sharp B_i^2)^r\right|\right|_1\leq\left|\left|\left(\left(\sum_{i=1}^mA_i\right)^{\frac{pr}{_2}}\left(\sum_{i=1}^mB_i\right)^{pr}\left(\sum_{i=1}^mA_i\right)^{\frac{rp}{_2}}\right)^{\frac{1}{p}}\right|\right|_1,$$for all $p>0$ and for all $r\geq1.$ We conjecture this inequality is also true for all unitarily invariant norms. We give an affirmative answer to the case of $m=2,$ $p\geq1$, $r\geq1$ and for all unitarily invariant norms. In other words, it is shown that $$\left|\left|\left|\left(A^{^2}\sharp B^{^2}\right)^{r}+\left(C^{^2}\sharp D^{^2}\right)^{r}\right|\right|\right|\leq \left|\left|\left|\left(\left(A+C\right)^{^\frac{rp}{_2}}\left(B+D\right)^{{rp}}\left(A+C\right)^{^\frac{rp}{_2}}\right)^{\frac{1}{_p}}\right|\right|\right|,$$for all unitarly invariant norms, for all $p\geq1$ and for all $r\geq1$, where $A,B,C,D$ are positive definite matrices. This gives an affirmative answer to the conjecture posed by Dinh, Ahsani and Tam in the case of $m=2$. The preceding inequalities directly lead to a recent result of Audenaert \cite{ANIFP}.