A sort of AM-GM inequality for matrices

It turns out that this question is non-trivial and was resolved only in 1999, by Rajendra Bhatia and Fuad Kittaneh in the paper Notes on matrix arithmetic–geometric mean inequalities, freely available here. They show that, for any unitarily invariant matrix norm $\|.\|$, $$ \| A B\|\leq \frac 14 \|(A+B)^2\|. $$ In the particular case of the Schatten norm with $p = 1$, this means $$ \sum_j s_j(AB) \leq \frac{1}{4} \sum_j s_j ((A + B)^2) = \frac{1}{4}\sum_j (s_j(A+B))^2, $$ which is the inequaity we were trying to show.

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Matrices

A.M. G.M. Inequality

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