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Recent research on Jensen's inequality for operators

Let f be an operator convex function defined on an interval I. Ch. Davis [1] proved1 a Schwarz inequality f (φ(x)) ≤ φ ( f (x)) (2) where φ : A → B(K) is a unital completely positive linear mapping from a C∗-algebra A to linear operators on a Hilbert space K, and x is a self-adjoint element in A with spectrum in I. Subsequently M. D. Choi [2] noted that it is enough to assume that φ is unital and positive. In fact, the restriction of φ to the commutative C∗-algebra generated by x is automatically completely positive by a theorem of Stinespring.