Kantorovich type operator inequalities via the Specht ratio

As a converse of the arithmetic and geometric mean inequality, Specht gave the ratio of the arithmetic one by the geometric one in 1960. We can reap the rich harvest of the Specht ratio in operator theory. In this paper, we shall present other characterizations of the chaotic order and the usual one associated with Kantorovich type inequalities via the Specht ratio. Among others, as an application of the grand Furuta inequality, we show that if A and B are positive operators and for some k 1, then A B is equivalent to where the Specht ratio Sk(r) is defined for each r>0 as