Convexity of a certain operator trace functional

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