  Relating to finding possible upper bounds for the probability of error for
discriminating between two quantum states, it is well-known that \begin{align*}
  \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big)
\end{align*} holds for every positive-valued matrix monotone function $f$,
where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. This study
demonstrates that the set of functions satisfying this inequality includes
additional elements and provides illustrative examples to support this claim.
Furthermore, we present a characterization of matrix decreasing functions based
on a matrix version of the above inequality.
