## 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
Full Text: