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].