A Numerical Radius Version of the Arithmetic-Geometric Mean of Operators

Alemeh Sheikhhosseini

In this paper, we obtain some numerical radius inequalities for operators, in particular for positive definite operators $A,B$ a numerical radius and some operator norm versions for arithmeticgeometric mean inequality are obtained, respectively as \[ w^2(Aharp B)eqslant\Big(\frac{A^2+B^2}2\Big)-\frac12ıf_{\|x\|=1}ẹlta(x) \] where $\delta(x)=\langle(A-B)x,x\rangle^2$, and \[ \|A\|\|B\|eqslant\frac12(\|A^2\|+\|B^2\|)-\frac12ıf_{\|x\|=\|y\|=1}ẹlta(x,y), \] where, $\delta(x,y)=(\langle Ay,y\rangle-\langle Bx,x\rangle)^2$.