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

ABSTRACT

In this paper, we first investigate some inequalities involving the p-weighted geometric operator mean $A{\mathrm{♯}}_{p}B=A^{1/2}(A^{-1/2}B{A}^{-1/2}{)}^p{A}^{1/2},$ 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 to $A{\mathrm{♯}}_{p}B$ with a high rate of convergence. Some relationships involving $A{\mathrm{♯}}_{p}B$ are deduced. Numerical examples illustrating the theoretical results are also discussed.

Keywords:

• Weighted geometric operator mean
• Heinz operator mean
• Tsallis relative operator entropy
• operator inequalities
• iterative algorithm
• convergence
• high rate of convergence

2010 Mathematics Subject Classifications:

• 46N10
• 46A20
• 47A63

Acknowledgments

The authors would like to express their sincere thanks to the two anonymous referees for their valuable comments and suggestions which allowed us to correct some mistakes and errors from the old version of the manuscript. Their useful remarks have substantially helped improve the quality of the final version of the present paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).