FURTHER INEQUALITIES RELATED TO SYNCHRONOUS AND ASYNCHRONOUS FUNCTIONS

Abstract

This paper intends to show some operator and norm inequalities involving synchronous and asynchronous functions. Among other inequalities, it is shown that if $A,B\in \mathcal B\left( \mathcal H \right)$ are two positive operators and $f,g:J\to \mathbb{R}$ are asynchronous functions, then
\[f\left( A \right)g\left( A \right)+f\left( B \right)g\left( B \right)\le \frac{1}{2}\left( {{f}^{2}}\left( A \right)+{{g}^{2}}\left( A \right)+{{f}^{2}}\left( B \right)+{{g}^{2}}\left( B \right) \right).\]