A proof of a conjectured determinantal inequality

The main goal of this paper is to prove the following determinantal inequality: det(A(k) + vertical bar B-s/2 A(s/2)vertical bar(2t/s)) <= (A(k) + A(t) B-t) <= det (A(k) + + vertical bar A(s/2) B-s/2 vertical bar(2t/s)) for any positive semi-definite matrices A and B, and for all 0 <= t <= s <= k. It generalizes several known determinantal inequalities, and one main consequence of it confirms Lin's conjecture which states that for positive semi-definite matrices A and B, det(A(2) + A(t) B-t) <= det(A(2) + vertical bar AB vertical bar(t)) for 0 <= t <= 2. We conclude with another related determinantal inequality. (C) 2020 Elsevier Inc. All rights reserved.