Simultaneous extensions of Selberg and Buzano inequalities

Abstract

We give a simultaneous extension of Selberg and Buzano inequalities: If $y1, y2$ and nonzero vectors $\{ z_i; i = 1, 2, \dots , n \}$ in a Hilbert space $\mathscr{H}$ satisfy the orthogonality condition $\langle y_k; z_i \rangle = 0$ for $i = 1, 2, \dots , n$ and $k = 1, 2,$ then \[ | \langle x, y_1 \rangle \langle x, y_2 \rangle | + \mathit{B} (y_1, y_2) \sum_i \frac{| \langle x, z_i \rangle |^2}{\sum_j | \langle z_i, Z-j \rangle |} \leq \mathit{B} (y_1, y_2) \|x\|^2 \] holds for all $x \in \mathscr{H}$, where $\mathit{B} (y1; y2) = \frac{1}{2} (\|y_1\| \|y_2\| + | \langle y_1, y_2 \rangle |)$. As an application, we discuss some refinements of the Heinz-Kato-Furuta inequality and the Bernstein inequality.