Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces (Q978431)

The author presents various reverses of the Jensen inequality. He proves that \(0 \leq \langle f(A)x,x\rangle - f(\langle Ax,x\rangle) \leq \langle f'(A)Ax,x\rangle - \langle Ax,x\rangle\), in which \(f: J \to \mathbb{R}\) is a convex and continuously differentiable function, \(A\) is a selfadjoint linear operator acting on a Hilbert space \(H\) with spectrum in \(J\), and \(x\in H\) is a unit vector. Applications for some particular convex functions are given as well.