Further results on the concavity of operator means

Abstract

In this work, we provide some further refinements for the following concavity inequalityAσB+CσD≤(A+C)σ(B+D) for positive invertible bounded linear operators A,B,C,D on a complex Hilbert space and for an operator mean σ. To this end, we use the operator majorization intended for comparing two n+1-tuples of pairs of operators, and prove a Hardy-Littlewood-Pólya-Karamata (HLPK) type theorem for the triangle map induced by σ. Next, we apply a specification of the HLPK Theorem for n=2 in order to obtain the required refinements.