Developed matrix inequalities via positive multilinear mappings

Utilizing the notion of positive multilinear mappings, we present some matrix inequalities. In particular, the Choi–Davis–Jensen inequality f(Φ(A,B))≤Φ(f(A),f(B))f(Φ(A,B))≤Φ(f(A),f(B)) does not hold in general for a matrix convex function f and a positive multilinear mapping Φ. We prove this inequality under certain condition on f. Moreover, some Kantorovich and convexity type inequalities including positive multilinear mappings are presented.