# On Operator Inequalities due to Ando-Kittaneh-Kosaki

### Jun Ichi Fuji

Osaka Kyoiku University, Japan### Masatoshi Fuji

Osaka Kyoiku University, Japan

## 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*.