The Hasegawa–Petz mean: Properties and inequalities

We study a family of means introduced by H. Hasegawa and D. Petz (1996) [13]. Properties with respect to the parameter, such as monotonicity and logarithmic concavity, further, monotonicity and concavity in the mean variables are shown. Besides, the comparison between the Hasegawa–Petz mean and the geometric mean is completely solved. The connection to earlier results on operator monotonicity and some applications are also discussed.