‎It is known that for every $t\\\\in (0,1]$ and every $k$-tuple of positive invertible operators ${\\\\Bbb A}=(A_1,\\\\ldots,A_k)$‎ , ‎the Ando--Hiai type inequality for operator power means‎ ‎\\\\[‎ ‎\\\\NORM{P_{\\\\frac{t}{r}}(\\\\omega; {\\\\Bbb A}^r)} \\\\leq \\\\NORM{P_t(\\\\omega; {\\\\Bbb A})^r} \\\\qquad \\\\mbox{for all $r\\\\geq 1$}‎ ‎\\\\]‎ ‎holds‎, ‎where $\\\\NORM{\\\\cdot}$ is the operator norm and $P_{t}(\\\\omega; {\\\\Bbb A})$ is the operator power mean‎. ‎However it is not known any relation between $P_t(\\\\omega; {\\\\Bbb A}^r)$ and $P_t(\\\\omega; {\\\\Bbb A})^r$ under the L\\\\\\\"{o}wner partial order‎. ‎In this paper‎, ‎we present some Ando--Hiai type inequalities for operator power means‎, ‎which give a relation between $P_t(\\\\omega;\\\\Phi(\\\\mathbb{A}^r))$ and $\\\\Phi\\\\left(P_t(\\\\omega;\\\\mathbb{A})^r \\\\right)$ for every unital positive linear map $\\\\Phi$‎. ‎In addition‎, ‎we obtain a difference counterpart to the information monotonicity‎.

Keywords