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 $[0,\infty )$ are discussed: Main inequality is $\Vert f(A)–f(B)\Vert \le \Vert f(|A–B|)\Vert$. 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 $\Vert f(A)–f(B)\Vert =f^{\prime }(t)\Vert A–B\Vert$ for $A,B\ge t>0$ and nonlinear $f$  if and only if $A=B$.