# Complementary Inequalities to Improved AM-GM Inequality

## TL;DR

This paper establishes new inequalities related to the AM-GM inequality for positive operators, providing bounds involving positive unital linear maps and operator means, extending classical inequalities.

## Contribution

It introduces complementary inequalities to the improved AM-GM inequality for positive operators, involving bounds with positive unital linear maps and operator means.

## Key findings

- Derived bounds for positive operators involving unital linear maps.
- Extended classical AM-GM inequalities with new operator bounds.
- Provided conditions under which these inequalities hold.

## Abstract

Following an idea of Lin, we prove that if $A$ and $B$ be two positive operators such that $0<mI\le A\le m'I\le M'I\le B\le MI$, then \begin{equation*} {{\Phi }^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{\Phi }^{2}}\left( A\#B \right), \end{equation*} and \begin{equation*} {{\Phi }^{2}}\left( \frac{A+B}{2} \right)\le \frac{{{K}^{2}}\left( h \right)}{{{\left( 1+\frac{{{\left( \log \frac{M'}{m'} \right)}^{2}}}{8} \right)}^{2}}}{{\left( \Phi \left( A \right)\#\Phi \left( B \right) \right)}^{2}}, \end{equation*} where $K\left( h \right)=\frac{{{\left( h+1 \right)}^{2}}}{4h}$ and $h=\frac{M}{m}$ and $\Phi $ is a positive unital linear map.