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In this paper, we first investigate some inequalities involving the  p -weighted geometric operator mean where  p  ∈ [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 towith a high rate of convergence. Some relationships involvingare deduced. Numerical examples illustrating the theoretical results are also discussed.

Further inequalities involving the weighted geometric operator mean and the Heinz operator mean