On a reverse of the Tan-Xie inequality for sector matrices and its applications

Abstract

In this short paper, we establish a reverse of the derived inequalities for sector matrices by Tan and Xie, with Kantorovich constant. Then, as application of our main theorem, some inequalities for determinant and unitarily invariant norm are presented.

Some Singular Value Inequalities for Sector Matrices Involving Operator Concave Functions

“…In eorem 2.22, letting f(X) � X for any X ∈ M n (C), we have a better bound ‖A!B‖ u ≤ sec3 (α)/2‖I n + A‖ u ‖I n + B‖ u .We close this paper with a determinant inequality.Compute|det(A♯B)| ≤ sec n (α)det(R(A♯B))(by Lemma 2.6 in [15]) ≤ sec 3n (α)det(R(A)♯R(B))(by Lemma 2.6) det I n + R(A) 􏼁 • det I n + R(B) 􏼁(by (12))Remark 2.25. We remark that eorem 2.24 is a refinement of Proposition 3.1 in[22].…”