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Modulus of convexity for operator convex functions

Quantum Physics 2015-06-17 v3 Mathematical Physics math.MP

Abstract

Given an operator convex function f(x)f(x), we obtain an operator-valued lower bound for cf(x)+(1c)f(y)f(cx+(1c)y)cf(x) + (1-c)f(y) - f(cx + (1-c)y), c[0,1]c \in [0,1]. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.

Cite

@article{arxiv.1310.0746,
  title  = {Modulus of convexity for operator convex functions},
  author = {Isaac H. Kim},
  journal= {arXiv preprint arXiv:1310.0746},
  year   = {2015}
}

Comments

5 pages, change of title. The new version shows that the main result of the original paper cannot be extended to convex functions that are not operator convex!

R2 v1 2026-06-22T01:39:07.558Z