Matrix power means and new characterizations of operator monotone functions

For positive definite matrices A and B, the Kubo-Ando matrix power mean is defined as P?(p,A,B)=A1/2(1+(A?1/2BA?1/2)p2)1/pA1/2(p?0). In this paper, for 0?p?1?q, we show that if one of following inequalities f(P?(p,A,B))?f(P?(1,A,B))?f(P?(q,A,B)) holds any then function f operator monotone on (0,?). We also study inverse problem non-Kubo-Ando means with powers 1/2 2. As a consequence, establish new charaterizations functions means.