Year: 2026
Type: preprint
Abstract: In this paper, we study noncommutative analogues of the classical Cauchy and Lagrange mean value theorems in the setting of free noncommutative function theory. For noncommutative functions on a uniformly open noncommutative set, we prove a Cauchy-type mean value inequality involving the right noncommutative difference operator $Δ_R$ and the G-derivative, and we establish a mean value theorem for noncommutative curves. Furthermore, we characterize uniform comparison (Lipschitz-type) inequalities at all matrix levels and prove a factorization theorem for noncommutative functions. The proofs combine the Hahn--Banach theorem, the fundamental theorem of calculus along a line segment, and the direct-sum compatibility of noncommutative functions.
Source: arXiv (Cornell University)
Authors: Du, Yali, Miao, Jun Jie, Tan, Tianjiao
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223948817