Variants of Ando–Hiai inequality for operator power means

Abstract

‎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<br>‎\\\\\\\\[‎<br>‎\\\\\\\\NORM{P_{\\\\\\\\frac{t}{r}}(\\\\\\\\omega; {\\\\\\\\Bbb A}^r)} \\\\\\\\leq \\\\\\\\NORM{P_t(\\\\\\\\omega; {\\\\\\\\Bbb A})^r} \\\\\\\\qquad \\\\\\\\mbox{for all $r\\\\\\\\geq 1$}‎<br>‎\\\\\\\\]‎<br>‎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.