When does Ando-Hiai inequality hold?

Let α be in (0,1) and r>0 and #α stand for the weighted operator geometric mean. We consider the following statement:A,B>0,A#αB≥I⇒Ar#αBr≥I. Ando and Hiai show that if r≥1, then this holds. In the present paper, we first prove the converse of this result, namely, the above statement holds only if r≥1. We next show that for each non-negative continuous function f on [0,∞) with f≤tr and f≠tr, there exist A,B>0 such that A#αB≥I and f(A)#αf(B)≱I. We also try to find a characterization of a continuous function f satisfyingA,B>0,A#αB≥I⇒f(A)#αf(B)≥I.