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The main goal of this paper is to prove the following determinantal inequality: det ⁡ (A k + | B s 2 A s 2 | 2 t s ) ≤ det ⁡ (A k + A t B t) ≤ det ⁡ (A k + | A s 2 B s 2 | 2 t 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 + | A B | t) for 0 ≤ t ≤ 2. We conclude with another related determinantal inequality. [ABSTRACT FROM AUTHOR]

In 2017, M. Lin formulated two conjectures concerning determinantal inequalities for positive semi-definite matrices A and B , and which can be stated as follows det ⁡ (A 2 + | A B | p) ≥ det ⁡ (A 2 + | B A | p) for p ≥ 0 and det ⁡ (A 2 + | A B | p) ≥ det ⁡ (A 2 + A p B p) for 0 ≤ p ≤ 2. The main goal of this paper is to confirm the first conjecture in a slightly more general setting namely in the case when A and B are Hermitian, and also to prove the second conjecture when 0 ≤ p ≤ 4 3. Various related inequalities are then presented and we conclude with an open log-majorization question. [ABSTRACT FROM AUTHOR]