Dragomir, S. S.; Mond, B. Some inequalities of Aczél type for Gramians in inner product spaces. (English) Zbl 0995.26008 Nonlinear Funct. Anal. Appl. 6, No. 3, 411-424 (2001). 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.]. Reviewer: János Aczél (Waterloo/Ontario) Cited in 6 Documents MSC: 26D15 Inequalities for sums, series and integrals 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) Keywords:inequalities for sums; Gramians; Hilbert space PDFBibTeX XMLCite \textit{S. S. Dragomir} and \textit{B. Mond}, Nonlinear Funct. Anal. Appl. 6, No. 3, 411--424 (2001; Zbl 0995.26008)