Remarks on a Question of Bourin for Positive Semidefinite Matrices

Let A and B be positive semidefinite matrices. For t is an element of[3/4,1] and for every unitarily invariant norm, it is shown that |||A(t)B(1-t)+B(t)A(1-t)|||<= 2(2(t-3/4))|||A+B|||and for t is an element of[0,1/4]|||A(t)B(1-t)+B(t)A(1-t)|||<= 2(2(1/4-t))|||A+B|||.These norm inequalities are sharper than an earlier norm inequality due to Alakhrass and closely related to an open question of Bourin. In fact, they lead to an affirmative solution of Bourin's question for t=1/4 and 3/4, which is a result due to Hayajneh and Kittaneh (Int J Math 32 (2150043):7, 2021).

Automated summary

For positive semidefinite matrices A and B, norm inequalities are proven for specific t values using unitarily invariant norms. These inequalities are sharper than previous ones derived by Alakhrass and closely related to an unsolved question posed by Bourin. In fact, these inequalities lead to an affirmative solution for Bourin's question at t=1/4 and 3/4, as demonstrated by Hayajneh and Kittaneh in 2021.