arXiv:1710.02937v5 [math.FA] 6 May 2020 NEW KANTOROVICH TYPE INEQUALITIES FOR NEGATIVE PARAMETERS SHIGERU FURUICHI AND HAMID REZA MORADI Abstract. We show the following result: Let A, B ∈ B (H) be two strictly positive operators such that A ≤ B and m1 H ≤ B ≤ M 1 H for some scalars 0 <m<M . Then B p ≤ exp  M 1 H − B M − m ln m p + B − m1 H M − m ln M p  ≤ K (m, M, p, q) A q for p ≤ 0, −1 ≤ q ≤ 0 where K (m, M, p, q) is the generalized Kantorovich constant with two parameters. In addition, we obtain Kantorovich type inequalities for the chaotic order. 1. Introduction and Preliminaries In what follows, a capital letter means a bounded linear operator on a complex Hilbert space H. An operator A is said to be positive (denoted by A ≥ 0 ) if 〈Ax, x〉≥ 0 for all x ∈H, and also an operator A is said to be strictly positive (denoted by A> 0) if A is positive and invertible. Here 1 H stands for the identity operator on H. Sp (A) denotes the usual spectrum of A. We take an interval I ⊆ R. If a positive function f : I → (0, ∞) satisfies (1.1) f ((1 − v) x + vy ) ≤ f 1−v (x)f v (y ), for all x, y ∈ I and v ∈ [0, 1], then we say that f is a logarithmically convex (or simply, log- convex) function on I . The weighted arithmetic-geometric mean inequality readily yields that every log-convex function is also convex. It is worth emphasizing that the function f (t)= t p is log-convex for p ≤ 0 on (0, ∞). The “L¨ owner-Heinz inequality” asserts that 0 ≤ A ≤ B ensures A p ≤ B p for any p ∈ [0, 1]. As is well-known, the L¨ owner-Heinz inequality does not always hold for p> 1. The following theorem due to Furuta [8, Theorem 2.1] (see also [9, Theorem 4.1]) is the starting point for our discussion. 2010 Mathematics Subject Classification. Primary 47A63, Secondary 46L05, 47A60. Key words and phrases. Kantorovich type inequality, chaotic order, order reversing operators, log-convex functions, Mond-Peˇ cari´ c method. 1