Inequalities related to Bourin and Heinz means with a complex parameter

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A, B   positive matrices, 0≤t≤10≤t≤1, and any unitarily invariant norm the following inequality holds⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀.⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t∈[14,34]. In this paper, using complex methods we extend this result to complex values of the parameter t=zt=z in the strip {z∈C:Re(z)∈[14,34]}. We give an elementary proof of the fact that equality holds for some z in the strip if and only if A and B   commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by sj(AtB1−t+BtA1−t)≤sj(A+B)sj(AtB1−t+BtA1−t)≤sj(A+B), answering in the negative a question made by K. Audenaert and F. Kittaneh. The methods of proof and examples can be adapted with no modifications to operator algebras (infinite dimensional setting), for instance it follows that the inequality above holds for Hilbert–Schmidt operators with their Banach algebra norm derived from the infinite trace of B(H)B(H).