Lieb type convexity for positive operator monotone decreasing functions

We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions certain concave functions, together strictly linear maps of matrices. This gives a partial generalization Hiai's recent work on power by allowing decreasing instead maps.Our proof is based variational formula the Legendre transform, strengthened in previous Kirihata second named author. also provide to framework unital tracial C -algebras Petz's work.