Abstract:
The Furuta inequality and its grand version are cornerstone results in operator theory, providing powerful
relationships between positive operators. While highly general, these classical inequalities are constrained
by a fixed set of parameters. This paper introduces a significant generalization by incorporating three new
parameters, θ, ϕ, and ψ. Our multi-parameter framework offers finer control over operator relationships,
enabling a wider range of applications and interpolations between known results. Specifically, for positive
operators A and B with 0 < m ≤ B ≤ M and h = M/m > 1, we establish the inequality:
A ≥ B ≥ 0 ⇒ A
α ≥
n
A
β
2
A
− θt
2 B
pA
− θt
2
s
A
β
2
o α
(p−t)s+β
,
where α = θ(1 − t + r) + ϕ and β = θr + ψ, for 0 ≤ t ≤ 1, p ≥ 1, s ≥ 1, r ≥ t, and θ, ϕ, ψ ≥ 0. We prove an
equivalent norm inequality and establish a reverse inequality using the generalized Kantorovich constant. As
applications, we derive new reverse forms of the Ando-Hiai inequality and demonstrate how our results unify
and extend classical inequalities, including those of L¨owner-Heinz and Araki-Cordes.
