Squaring operator P\'{o}lya--Szeg\"{o} and Diaz--Metcalf type inequalities

Abstract

We square operator P\'{o}lya--Szeg\"{o} and Diaz--Metcalf type inequalities as follows: If operator inequalities $0<m_{1}^{2} \leq A\leq M_{1}^{2}$ and $0<m_{2}^{2}\leq B\leq M_{2}^{2}$ hold for some positive real numbers $m_{1}\leq M_{1}$ and $m_{2}\leq M_{2}$, then for every unital positive linear map $\Phi$ the following inequalities hold: \begin{eqnarray*} (\Phi(A)\sharp\Phi(B))^2 &\leq&\left(\frac{M_1M_2 + m_1m_2}{2\sqrt{M_1M_2m_1m_2}}\right)^4\Phi(A\sharp B)^{2} \end{eqnarray*} and \begin{eqnarray*} \left( \frac{M_2m_2}{M_1m_1}\Phi (A) + \Phi (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)^2\Phi (A\sharp B)^2\,. \end{eqnarray*}