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