Convexity of a certain operator trace functional

In this article the operator trace function Λr,s(A)[K,M]:=tr(K⁎ArMArK)s is introduced and its convexity concavity properties are investigated. This has a direct connection to several well-studied functions that appear in quantum information theory, particular when studying data processing inequalities of various relative entropies. paper interplay between Λr,s well-known Γp,s Ψp,q,s used study stability their (concavity) properties. may be ensure convex (concave) certain parameter ranges M=I or K=I. However, our main result shows surprisingly lost perturbing those matrices even little. To complement theorem, domain Λ itself examined. The final states never concave it if only r=1 s≥1/2.