## Nihonkai Mathematical Journal

### 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.

#### Article information

Source
Nihonkai Math. J., Volume 25, Number 1 (2014), 45-63.

Dates
First available in Project Euclid: 17 October 2014

https://projecteuclid.org/euclid.nihmj/1413555412

Mathematical Reviews number (MathSciNet)
MR3270971

Zentralblatt MATH identifier
1311.47020

Subjects
Primary: 47A63: Operator inequalities

#### Citation

Fujii, Masatoshi; Matsumoto, Akemi; Tominaga, Masaru. Simultaneous extensions of Selberg and Buzano inequalities. Nihonkai Math. J. 25 (2014), no. 1, 45--63. https://projecteuclid.org/euclid.nihmj/1413555412

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