A complementary result of Kantorovich type order preserving inequalities by Mićić–Pečarić–Seo

Abstract

We shall discuss operator inequalities which are obtained by elementary lemma (Lemma 3.1), associated with Hölder–McCarthy and Kantorovich inequalities. Firstly we shall give the following complementary result to Mićić et al. [Linear Algebra Appl., 360 (2003) 15]. Let A and B be two strictly positive operators on a Hilbert space H such that M 1 I⩾ A⩾ m 1 I>0 and M 2 I⩾ B⩾ m 2 I>0, where M 1> m 1>0, M 2> m 2>0 and A⩾ B. (a) If p>1 and q>1, then the following inequality holds: (q−1) q−1 q q (M 2 p−m 2 p) q (M 2−m 2)(m 2M 2 p−M 2m 2 p) q−1 A q⩾B p for m 2 p−1q⩽ M 2 p−m 2 p M 2−m 2 ⩽M 2 p−1q. (b) If p<0 and q<0, then the following inequality holds: (m 1M 1 p−M 1m 1 p) (q−1)(M 1−m 1) (q−1)(M 1 p−m 1 p) q(m 1M 1 p−M 1m 1 p) qB q⩾A p for m 1 p−1q⩽ M 1 p−m 1 p M 1−m 1 ⩽M 1 p−1q. We remark that (a) is shown in [Linear Algebra Appl., 360 (2003) 15] as an extension of two variable version of our previous one variable one [J. Inequal. Appl. 2 (1998) 137]. Secondly, we shall show the following extension of two parameters type of an extension of Fujii et al. [Sci. Math. 1 (1998) 307] on the determinant of an operator. Let T be strictly positive operators on a Hilbert space H such that MI⩾ T⩾ mI>0. Then the following inequality holds: S h(p,q)Δ x(T q)⩾(T px,x)⩾Δ x(T p) for p>0 and q>0, where S h ( p, q) is defined by S h(p,q)=m p−q h q/(h p−1) e logh q/(h p−1) if q⩽ h p−1 logh ⩽qh p andthe determinant Δ x ( T) for strictly positive operator T at a unit vector x in Hilbert space H is defined by Δ x(T)= exp〈(( logT)x,x)〉. As an application of this result, we shall give an alternative proof of two variable version of characterization of the chaotic order in [Linear Algebra Appl. ibid., Theorem 4.4].