Properties And Inequalities About Sector Matrices

We mainly study the properties of sector matrices,several inequalities of concave functions on sector matrices and some inequalities about sector matrices and geometric mean.First,we prove the numerical ranges of the conjugate,inverse and Schur complement of any order of are in the same .The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product,star-congruent matrix and unitary matrix of polar decompostion are also included in the same sector.Furthermore,we extend some inequalities about eigenvalues and singular values and the linear fractional maps to this class of matrices.Then,we mainly prove two inequalities for concave functions and partitioned sector matrices.These complement some results of Zhang [52].Finally,we extend some known determinantal inequalities of positive matrices to a larger class of sector matrices.Besides,we also give an upper bound for the geometric mean defined by Drury [13].