An extension of the Lowner-Heinz inequality

We extend the celebrated L"owner–Heinz inequality by showing that if $A,B$ are Hilbert space operators such that $A>B\ge 0$, then A^r - B^r \geq ||A||^r-(||A||- \frac{1}{||(A-B)^{-1}||})^r > 0 for each $0<r\le 1$. As an application we prove that \log A - \log B \geq \log||A||- \log(||A||-\frac{1}{||(A-B)^{-1}||})>0.