LAUSR

In this note, we prove that some of recent Rotfel’d type inequalities are equivalent, which is an extension of Huang, Wang and Zhang [Linear Multilinear Algebra 66 (2018) 1626–1632]. Among other results, it is shown that if f : [0,∞)→ [0,∞) is a concave function and A ∈M2(Mn) is a normal matrix with its numerical range contained in a sector: Sα = {z ∈ C : Re z 3⁄4 0, |Im z|¶ (Re z) tanα} for some α ∈ [0, π 2 ), then ‖ f (|A|)‖¶ 2 f secα 2 |A11 +A22| for any unitarily invariant norm ‖·‖. This inequality improves a recent result of Zhao and Ni [Linear Multilinear Algebra 66 (2018) 410–417].