Inequalities for sector matrices with negative power

In this paper, we present some inequalities for sector matrices with negative power. Among other results, we prove that if A,B ? Mn(C) with W(A),W(B) ? S?, then for any positive unital linear map ?, it holds R((1-v)?(A) + v?(B))r ? cos2r(?)R?((1-v)Ar + vBr), where v ? [0,1] and r ? [-1,0]. This improves Tan and Xie?s Theorem 2.4 in [22] if setting ?(X) = X for every X ? Mn(C) and replacing A by A-1, B by B-1, respectively, and r=-1, which is also a special result of Bedrani, Kittaneh and Sababheh?s Theorem 4.1 in [4].