It is known that the power function f (t) = t 2 is not matrix monotone. Recently, it has been shown that t 2 preserves the order in some matrix inequalities. We prove that if $A = (A_1 ,\cdots , A_k)$ and $B = (B_1 , \cdots , B_k)$ are k-tuples of positive matrices with 0 < m ≤ A i , B i ≤ M (i = 1,. .. , k) for some positive real numbers m < M, then $$\Phi^2(A_1^{-1},\cdots,A_k^{-1})e \bigl(\frac{(1+v^k)^2}{4v^k}\bigr)^2 \Phi^{-2}(A_1,\cdots,A_k)$$ and $$\Phi^2(\frac{A_1+B_1}{2},\cdots,\frac{A_k+B_k}{2})e \bigl(\frac{(1+v^k)^2}{4v^k}\bigr)^2 \Phi^2(A_1\#B_1,\cdots A_k\#B_k)$$ where Φ is a unital positive multilinear mapping and v = M m is the condition number of each A i .