SOME INEQUALITIES INVOLVING OPERATOR MONOTONE FUNCTIONS AND OPERATOR MEANS

In this paper we show that if f : [0,infinity) -> [0,infinity) is an operator monotone function and A, B are positive operators such that 0 < pA <= B <= qA, then for all a is an element of [0,1] f(A) (alpha)f (B) <= max {S(p), S(q)}f(A(alpha)B), where S(t) is the so called Specht's ratio, and (alpha) is alpha-geometric mean. Moreover, we present some majorization and norm inequalities for operator monotone functions. Operator monotone decreasing functions are also discussed.