Some integral inequalities for operator monotonic functions on Hilbert spaces

Abstract

Abstract Let f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I, 0≤∫01p(t)f((1-t)A+tB)dt-∫01p(t)dt∫01f((1-t)A+tB)dt≤14[ p(1)-p(0) ][ f(B)-f(A) ] \matrix{0 \hfill & { \le \int\limits_0^1 {p\left( t \right)f\left( {\left( {1 - t} \right)A + tB} \right)dt - \int\limits_0^1 {p\left( t \right)dt\int\limits_0^1 {f\left( {\left( {1 - t} \right)A + tB} \right)dt} } } } \hfill \cr {} \hfill & { \le {1 \over 4}\left[ {p\left( 1 \right) - p\left( 0 \right)} \right]\left[ {f\left( B \right) - f\left( A \right)} \right]} \hfill \cr } in the operator order. Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.