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

TL;DR

This paper extends classical Pólya--Szegő and Diaz--Metcalf inequalities to operator settings by establishing squared inequalities under bounded operator conditions, involving positive linear maps and operator means.

Contribution

The paper introduces new squared operator inequalities of Pólya--Szegő and Diaz--Metcalf types, generalizing existing inequalities with bounds involving positive linear maps.

Findings

Derived squared inequalities for operator means under bounded conditions

Established bounds involving positive linear maps and operator inequalities

Extended classical inequalities to a broader operator framework

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*}