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Some inequalities of Aczél type for Gramians in inner product spaces. (English) Zbl 0995.26008

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 \[ \left(a-\prod_{j=1}^k\|x_k\|^4\right)\left(c-\prod_{j=k+1}^m\|x_k\|^4\right)\leq (b\pm\Gamma(x_1,...,x_m))^2\quad (k=1,...,m) \] for all \(x_j\in H (j=1,...,m\geq 2)\) with \(\prod_{j=1}^k\|x_k\|^4\leq a\) or \(\prod_{j=k+1}^m\|x_k\|^4\leq c.\)
[Remark: There seems to be a misprint in Theorem 2, because the first and second item after “\(\min\)” appear to be identical.].

MSC:

26D15 Inequalities for sums, series and integrals
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
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