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

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 arithmetic-geometric mean inequality are obtained, respectively as ω 2 ( A ♯ B ) ⩽ ω ( A 2 + B 2 2 ) − 1 2 inf | | x | | = 1 δ ( x ) , where δ(x) = 〈(A - B)x,x〈2 , and | | A | | | | B | | ⩽ 1 2 ( | | A 2 | | + | | B 2 | | ) − 1 2 inf | | x | | = | | y | | = 1 δ ( x , y ) , where, δ(x, y) = (⟨Ay, y⟩ – ⟨Bx, x⟩)2 :