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Let A,B and C be three positive definite matrices. In this paper, we show a characterization of A B C via the Karcher mean as follows, Λ(ω ;A−p1 ,B−p2 ,B sC3 ) C, Λ(ω ;C−p1 ,B−p2 ,Bt sA3) A hold for t ∈ [0,1] , s 1 , p1, p2 > 0 and p3 > 1 , where ω̂ = ( 1 p1+1 , 1 p2+1 , 2 (p3−t)s+t−1 ) , ω = (w1,w2,w3) = ω̂ ‖ω̂‖1 .

A characterization of matrix inequality A≥B≥C via Karcher mean