A complement of the Ando–Hiai inequality

In this paper, we present a complement of a generalized Ando–Hiai inequality due to Fujii and Kamei [M. Fujii, E. Kamei, Ando–Hiai inequality and Furuta inequality, Linear Algebra Appl. 416 (2006) 541–545]. Let AA and BB be positive operators on a Hilbert space HH such that 00h>0 asK(h,p)≔hp-h(p-1)(h-1)p-1php-1hp-hpfor all real numbers p∈Rp∈R.