On Some General Inequalities Related to Operator AM-GM Inequality

Abstract

We present some operator AM-GM inequalities that generalize and refine the
derived results in the literature. Among other results, it is shown that if A, B ∈ B (H) such
that mI ≤ A, B ≤ MI for some scalars 0 < m < M, then for any p ≥ 1 and 0 ≤ v ≤ 1,
(A∇vB)
p ≤
(
m∇λM
m♯λM
)p
(A♯vB)
p + pβI,
where λ = min {v, 1 − v} and
β = sup
x∈H
∥x∥=1
{
⟨(A∇vB)
p
x, x⟩ − ⟨A∇vBx, x⟩
⟨
(A∇vB)
p−1
x, x⟩} .