Squaring a reverse AM-GM inequality
Volume 215 / 2013
Studia Mathematica 215 (2013), 187-194
MSC: Primary 47A63; Secondary 47A30
DOI: 10.4064/sm215-2-6
Abstract
Let $A , B$ be positive operators on a Hilbert space with $0 < m \le A, B \le M$. Then for every unital positive linear map $\varPhi $, $$ \varPhi ^2((A+B)/2)\le K^2(h)\varPhi ^{2}(A\mathbin {\sharp } B), $$ and $$ \varPhi ^2((A+B)/2)\le K^2(h)(\varPhi (A)\mathbin {\sharp } \varPhi (B))^{2}, $$ where $A\mathbin {\sharp } B$ is the geometric mean and $K(h)={(h+1)^2/(4h)}$ with $h=M/m$.
