On Operator Inequalities due to Ando–Kittaneh–Kosaki

Operator norm inequalities due to Ando–Kittaneh–Kosaki for positive operators A, B and a non-negative operator monotone function f on [0,∞) are discussed: Main inequality is \Vert f(A) – f (B)\Vert ≤ \Vert f(|A–B|)\Vert . It is shown that the equality holds for invertible A, B and non-linear f if and only if A = B and f(0) = 0 . Similarly, from the Kittaneh–Kosaki inequality, we show that  \Vert f(A) – f(B)\Vert = f'(t)\Vert A–B\Vert for A, B≥ t> 0 and nonlinear f  if and only if A = B .