Some inequalities for operator monotone functions

In this paper we show that, if that the function f : [0, ∞) → 𝔾 is operator monotone in [0, ∞) then there exist b ≥ 0 and a positive measure m on [0, ∞) such that [ f(B)-f(A) ](B-A)==b(B-A)2+∫0∞s2[ ∫01[ ((1-t)A+tB+s)-1(B-A) ]2dt ]dm(s) \matrix{ {\left[ {f\left( B \right) - f\left( A \right)} \right]\left( {B - A} \right) = } \hfill \cr { = b{{\left( {B - A} \right)}^2} + \int_0^\infty {{s^2}\left[ {\int_0^1 {{{\left[ {{{\left( {\left( {1 - t} \right)A + tB + s} \right)}^{ - 1}}\left( {B - A} \right)} \right]}^2}dt} } \right]dm\left( s \right)} } \hfill \cr } for all A, B > 0. Some necessary and sufficient conditions for the operators A, B > 0 such that the inequality f(B)B+f(A)A≥f(A)B+f(B)A f\left( B \right)B + f\left( A \right)A \ge f\left( A \right)B + f\left( B \right)A holds for any operator monotone function f on [0, ∞) are also given.