An estimation of quasi-arithmetic mean by arithmetic mean and its applications (Q874871)

For an increasing, strictly convex function \(f\) on an interval \(I\) and a selfadjoint operator \(A\) on a Hilbert space \(H\) whose spectrum is contained in \(I\), \(f^{-1}(\langle f(A)x, x\rangle)\geq \langle Ax,x\rangle\) is called the quasi-arithmetic mean inequality. In this paper, for each \(\lambda>0\), the author gives upper bounds for \[ f^{-1}(\langle f(A)x, x\rangle)-\lambda \langle Ax,x\rangle. \] Then the author derives some basic inequalities by looking at specific choices for \(f(t)\), such as \(t^p\), \(e^t\) and \(t\log t\).