Characterization of the monotonicity by the inequality

Let $\phi$ be a normal state on the algebra $B(H)$ of all bounded operators on a Hilbert space $H$, $f$ a strictly positive, continuous function on $(0,\mathrm{\infty })$, and let $g$ be a function on $(0,\mathrm{\infty })$ defined by $g(t) = \frac{t}{f(t)}$. We will give characterizations of matrix and operator monotonicity by the following generalized Powers-St\ormer inequality: $$\phi (A+B)-\phi (\mathrm{\mid }A-B\mathrm{\mid })\le 2\phi (f(A{)}^1\mathrm{/}2g(B)f(A{)}^1\mathrm{/}2),$$ whenever $A,B$ are positive invertible operators in $B(H)\mathrm{.}$