Generalized Multi-Parameter Furuta Inequality and Its Reverse

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.