The Schur complement inequalities involving sectorial matrices

In this paper, we obtain several new matrix inequalities that connect the Schur complement of sectorial matrices, involving a matrix monotone function defined by the harmonic mean integral. Among other results, it is shown that, if A and B are two matrices such that W(A),W(B)⊆Sα for some α∈[0,π2), and σf is a matrix mean, then ℜ(S(A)σfS(B))≤(sec⁡α)4ℜ(S(AσfB)). Moreover, it is obtained ℜ(f(S(A)))≤(sec⁡α)4ℜ(S(f(A))), where f is a non-negative matrix monotone function with f(1)=1 and A is a matrix such that W(A)⊆Sα for some α∈[0,π2). Additionally, we present some assertions related to matrix monotone functions.