Spechtʹs ratio and complementary inequalities to Golden–Thompson type inequalities on the Hadamard product Original Research Article

As a converse of the arithmetic–geometric mean inequality, W. Specht [Math. Z. 74 (1960) 91–98] estimated the ratio of the arithmetic mean to the geometric one. In this paper, we shall show complementary inequalities to the matricial generalization of Oppenheimʹs inequality and the Golden–Thompson type inequalities on the Hadamard product by T. Ando [Linear Algebra Appl. 26 (1979) 203; Linear Algebra Appl. 241–243 (1996) 105], in which Spechtʹs ratio plays an important role. As an application, we shall obtain a complementary inequality to the Hadamard determinant inequality.