Squaring operator Pólya--Szegö and Diaz--Metcalf type inequalities

Abstract

We square operator Pólya--Szegö and Diaz--Metcalf type inequalities as follows: If operator inequalities $0<m_1^2\le A\le M_1^2$ and $0<m_2^2\le B\le M_2^2$ hold for some positive real numbers $m_1\le M_1$ and $m_2\le M_2$, then for every unital positive linear map $\hat{I}¦$ the following inequalities hold: \begin{eqnarray*} (Φ(A)\sharpΦ(B))^2 &\leq&\left(\frac{M_1M_2 + m_1m_2}{2\sqrt{M_1M_2m_1m_2}}\right)^4Φ(A\sharp B)^{2} \end{eqnarray*} and \begin{eqnarray*} \left( \frac{M_2m_2}{M_1m_1}Φ(A) + Φ(B) \right)^2 \leq \left( \frac{(M_1m_1(M_2^2 + m_2^2) + M_2m_2(M_1^2 + m_1^2))^2}{8\sqrt{M_2M_1m_1m_2} M_1^2m_1^2M_2m_2} \right)^2Φ(A\sharp B)^2\,. \end{eqnarray*}

10 pages, to appear in Linear Algebra Appl. (LAA)