Simultaneous extensions of Diaz-Metcalf and Buzano inequalities

Abstract

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