Characterization of the monotonicity by the inequality

Let $φ$ 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 $(0, \infty)$, and let $g$ be a function on $(0, \infty)$ 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: $φ(A + B) - φ(|A - B|) \leq 2φ(f(A)^1/2g(B)f(A)^1/2),$ whenever $A, B$ are positive invertible operators in $B(H).$

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