Simultaneous extensions of Selberg and Buzano inequalities (Q466171)

Let \(H\) be a Hilbert space. The Selberg inequality states that, for given nonzero vectors \(\{z_i: i = 1, 2, \dots, n\}\) in \(H\), \(\sum_i\frac{|\langle x,z_i\rangle|}{\sum_j|\langle z_i,z_j\rangle|} \leq \|x\|^2\) holds for any \(x \in H\). On the other hand, the Buzano inequality says that \(|\langle x,y_1\rangle\langle x,y_2\rangle| \leq \frac{1}{2}(\|y_1\|\,\|y_2\|+|\langle y_1,y_2\rangle|)\|x\|^2\) is valid for all \(x, y_1, y_2 \in H\), see \textit{M. Khosravi} et al. [Filomat 26, No. 4, 827--832 (2012; Zbl 1289.47036)]. In the paper under review, the authors present a simultaneous extension of both the Selberg and Buzano inequalities by showing that, if the set \(\{y_1, y_2\}\) is orthogonal to the set \(\{z_i: i = 1, 2, \dots, n\}\) of nonzero vectors in a Hilbert space \(H\), then NEWLINE\[NEWLINE|\langle x,y_1\rangle\langle x,y_2\rangle|+B(y_1,y_2)\sum_i\frac{|\langle x,z_i\rangle|}{\sum_j|\langle z_i,z_j\rangle|} \leq B(y_1,y_2)\|x\|^2NEWLINE\]NEWLINE for all \(x \in H\), where \(B(y_1,y_2)=\frac{1}{2}(\|y_1\|\,\|y_2\|+|\langle y_1,y_2\rangle|)\). As application, they give some refinements of the Heinz-Kato-Furuta inequality and the Bernstein inequality.