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- W2903810109 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 $0 le r_j le frac{mathop{mathrm{Re}}{leftlangle{x_i},{z_j}rightrangle}}{left|{x_i}right|}$ for $i=1,ldots,n$. If $y_1,y_2 in mathscr{H}$ satisfy ${leftlangle{y_k},{z_j}rightrangle}=0$ for $k=1,2$ and $j=1,ldots,m$, then $${left|{leftlangle{sum x_i},{y_1}rightrangle} {leftlangle{sum x_i},{y_2}rightrangle}right|} + left(sum frac{r_j^2}{c_j}right) left(sum {left|{x_i}right|}right)^2 mathcal{B}left({y_1}{y_2}right) le mathcal{B}left({y_1}{y_2}right) left|{sum {x_i}}right|^2,$$ where $mathcal{B}left({y_1},{y_2}right) :=frac12(left|{y_1}right| left|{y_2}right| +{left|{leftlangle{y_1},{y_2}rightrangle}right|})$ and $c_j = sum_h|{leftlangle{z_h},{z_j}rightrangle}|$ for $j=1, ldots, m$." @default.
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- W2903810109 date "2016-01-01" @default.
- W2903810109 modified "2023-09-23" @default.
- W2903810109 title "Simultaneous extensions of Diaz-Metcalf and Buzano inequalities" @default.
- W2903810109 hasPublicationYear "2016" @default.
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