ESTIMATIONS OF HERON MEANS FOR POSITIVE OPERATORS

The arithmetic-geometric mean inequality induces the path of Heron means through these two means by H-r(mu) (A,B) = r(A#B-mu)+(1-r)(A del B-mu) for each mu is an element of [0,1], r is an element of R and positive operators A, B on a Hilbert space. In this note, we estimate H-r(mu) (A, B) by the harmonic mean. As an application of this method, we refine the arithmetic-geometric mean inequality under the assumption of the strict order A-B >= m > 0.