A generalization of the Powers-Størmer-Ogata inequality

We show that for any positive operator monotone function $f$ the inequality ${\int }_{[0,\mathrm{\infty })}f(t)\mathrm{d}{\parallel E^{{\mathrm{\Delta }}_{\phi ,\psi }}(t){\xi }_{\psi }\parallel }^2+f_{\mathrm{\infty }}^{\mathrm{\prime }}\phi (1-s(\psi ))\ge \frac{f(1)}{2}(\phi (1)+\psi (1)-\parallel \phi -\psi \parallel )$ holds, where $\phi ,\psi \in {\mathcal{M}}_{\ast ,+}$ are two normal positive linear functionals on a von Neumann algebra $\mathcal{M}$, and ${\mathrm{\Delta }}_{\phi ,\psi }$ is the associated relative modular operator. Choosing $f(t)=t^s$ with $s\in [0,1]$, the Powers-Størmer-Ogata inequality is recovered.