Some inequalities of Aczél type for Gramians in inner product spaces (Q2778302)

The following is one of the technically simpler among the results offered. Let \(H\) be an inner product space over the field of real or complex numbers, \(a, c\) be positive numbers, \(b^2\geq ac,\) and \(\Gamma\) denote the Gramian. Then NEWLINE\[NEWLINE\left(a-\prod_{j=1}^k\|x_k\|^4\right)\left(c-\prod_{j=k+1}^m\|x_k\|^4\right)\leq (b\pm\Gamma(x_1,...,x_m))^2\quad (k=1,...,m)NEWLINE\]NEWLINE for all \(x_j\in H (j=1,...,m\geq 2)\) with \(\prod_{j=1}^k\|x_k\|^4\leq a\) or \(\prod_{j=k+1}^m\|x_k\|^4\leq c.\)NEWLINENEWLINENEWLINE[Remark: There seems to be a misprint in Theorem 2, because the first and second item after ``\(\min\)'' appear to be identical.].