Abstract
In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean
A#B-p = A(1/2)(A(-1/2)BA(-1/2))(p) A(1/2),
where p is an element of [0, 1] is a real number and A, B are two positive invertible operators acting on a Hilbert space. As applications, we obtain some inequalities about the so-called Tsallis relative operator entropy. We also give some inequalities involving the Heinz operator mean. Our results refine some inequalities existing in the literature. In a second part, we construct iterative algorithms converging to A#B-p with a high rate of convergence. Some relationships involving A#B-p are deduced. Numerical examples illustrating the theoretical results are also discussed.