Solutions to some open problems in matrix analysis

This paper deals with several open questions in matrix analysis. We answer a question of Audenaert and Kittaneh by strengthening the Aujla–Bourin subadditivity inequality. We also answer questions posed by Bourin and the second author: (1) we extend the reciprocal Lie–Trotter theorem of Audenaert and Hiai to a product involving a third matrix; (2) we determine the best constants in inequalities for sums of contractions and positive block matrices, including the triangle inequality $$\mathrm{\mid }A+B+C\mathrm{\mid }\le \frac{3}{4}I+\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }+\mathrm{\mid }C\mathrm{\mid }$$ for three contractions $A,B,C$, where $3\mathrm{/}4$ cannot be replaced by a smaller constant; and (3) we prove an eigenvalue inequality involving the symmetric modulus $(\mathrm{\mid }X\mathrm{\mid }+\mathrm{\mid }{X}^{\ast }\mathrm{\mid })\mathrm{/}2$ studied by Bourin, Lee, and Zhang.