A matrix inequality including that of Kantorovich-Hermite. II (Q1206859)

[For part I see the second author, ibid. 13, 49-52 (1966; Zbl 0131.015).] For an \(n\times n\) positive definite Hermitian matrix \(A\) the authors prove that \[ 1\geq{(A^{r+s}x,x)\over (A^{sp}x,x)^{1/p}(A^{qr}x,x)^{1/q}}\geq K \] for the real numbers \(r,s,p,q\), with \(1/p+1/q=1\), \((ps-rq)/p>0\), \(p>1\), where \(K\) is a certain number. For \(p<1\) \((\neq 0)\) the reverse inequalities hold. Specializing \(r,s,p,q\) one obtains several classical inequalities.