On Operator Inequalities due to Ando-Kittaneh-Kosaki

Abstract

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 ||f (A) – f (B)|| ≤ ||f(|A–B|)||. 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 ||f(A) – f(B)||  = f''(t)||A–B|| for A, B ≥ t> 0 and nonlinear f  if and only if A = B.