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Simultaneous extensions of Diaz-Metcalf and Buzano inequalities. (English) Zbl 06820444

Summary: We give a simultaneous extension of Diaz-Metcalf and Buzano inequalities: Let \(z_1,\dots,z_m\) be nonzero vectors in a Hilbert space \(\mathscr{H}\). Suppose that \(x_1,\dots,x_n \in \mathscr{H}\) satisfy that for each \(j=1,\dots,m\) there exists a constant \(r_j\) such that \(0 \leq r_j \leq \frac{\operatorname{Re}{\langle{x_i},{z_j}\rangle}}{\|{x_i}\|}\) for \(i=1,\dots,n\). If \(y_1,y_2 \in \mathscr{H}\) satisfy \({\langle{y_k},{z_j}\rangle}=0\) for \(k=1,2\) and \(j=1,\dots,m\), then \[ {\left|{\left\langle{\sum x_i},{y_1}\right\rangle} {\left\langle{\sum x_i},{y_2}\right\rangle}\right|} + \left(\sum \frac{r_j^2}{c_j}\right) \left(\sum {\left\|{x_i}\right\|}\right)^2 \mathcal{B}\left({y_1}{y_2}\right) \leq \mathcal{B}\left({y_1}{y_2}\right) \left\|{\sum {x_i}}\right\|^2, \] where \(\mathcal{B}({y_1},{y_2}) :=\frac12(\|{y_1}\| \|{y_2}\| +{|{\langle{y_1},{y_2}\rangle}|})\) and \(c_j = \sum_h|{\langle{z_h},{z_j}\rangle}|\) for \(j=1,\dots, m\).

MSC:

47A63 Linear operator inequalities
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Full Text: Euclid