Nihonkai Mathematical Journal

Simultaneous extensions of Diaz-Metcalf and Buzano inequalities

Masatoshi Fujii, Akemi Matsumoto, and Masaru Tominaga

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 $j=1, \ldots, m$.

Article information

Source
Nihonkai Math. J., Volume 27, Number 1-2 (2016), 17-27.

Dates
Received: 20 November 2015
Revised: 23 May 2016
First available in Project Euclid: 14 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1505419738

Mathematical Reviews number (MathSciNet)
MR3698238

Zentralblatt MATH identifier
06820444

Subjects
Primary: 47A63: Operator inequalities

Keywords
Diaz-Metcalf inequality Selberg inequality Buzano inequality Furuta inequality Heinz-Kato-Furuta inequality and chaotic order

Citation

Fujii, Masatoshi; Matsumoto, Akemi; Tominaga, Masaru. Simultaneous extensions of Diaz-Metcalf and Buzano inequalities. Nihonkai Math. J. 27 (2016), no. 1-2, 17--27. https://projecteuclid.org/euclid.nihmj/1505419738


Export citation

References

  • M. L. Buzano, Generalizzazione della diseguaglianza di Cauchy-Schwarz, Rend. Sem. Mat. Univ. e Politech. Torino 31 (1971/73), 405–409 (1974).
  • J. B. Diaz and F. T. Metcalf, A complementary triangle inequality in Hilbert and Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 88–97.
  • M. Fujii, Furuta's inequality and its mean theoretic approach, J. Operator Theory 23 (1990), 67–72.
  • M. Fujii, T. Furuta and E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Algebra Appl. 179 (1993), 161–169.
  • M. Fujii, J.-F. Jiang and E. Kamei, Characterization of chaotic order and its application to Furuta inequality, Proc. Amer. Math. Soc. 125 (1997), 3655–3658.
  • M. Fujii, J.-F. Jiang, E. Kamei and K. Tanahashi, A characterization of chaotic order and a problem, J. Inequal. Appl. 2 (1998), 149–156.
  • M. Fujii and E. Kamei, Furuta's inequality and a generalization of Ando's theorem, Proc. Amer. Math. Soc. 115 (1992), 409–413.
  • M. Fujii, A. Matsumoto and M. Tominaga, Simultaneous extensions of Selberg and Buzano inequalities, Nihonkai Math. J. 25 (2014), 45–63.
  • M. Fujii and R. Nakamoto, Simultaneous extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math. J. 9 (1998), 219–225.
  • M. Fujii and R. Nakamoto, Extensions of Heinz-Kato-Furuta inequality, Proc. Amer. Math. Soc. 128 (2000), 223–228.
  • M. Fujii and R. Nakamoto, Extensions of Heinz-Kato-Furuta inequality, II, J. Inequal. Appl. 3 (1999), 293–302.
  • M. Fujii and H. Yamada, Around the Bessel inequality, Math. Japon. 37 (1992), 979–983.
  • T. Furuta, $A \ge B \ge 0$ assures $(B^rA^pB^r)^{1/q} \ge B^{(p+2r)/q}$ for $r \ge 0, p \ge 0, q \ge 1$ with $(1+2r)q \ge p+2r$, Proc. Amer. Math. Soc. 101 (1987), 85–88.
  • T. Furuta, An elementary proof of an order preserving inequality, Proc. Japan Acad. 65 (1989), 126.
  • T. Furuta, An extension of the Heinz-Kato theorem, Proc. Amer. Math. Soc. 120 (1994), 785–787.
  • T. Furuta, Determinant type generalizations of the Heinz-Kato theorem via the Furuta inequality, Proc. Amer. Math. Soc. 120 (1994), 223–231.
  • E. Kamei, A satellite to Furuta's inequality, Math. Japon. 33 (1988), 883–886.
  • K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc. 124 (1996), 141–146.
  • M. Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl. 2 (1999), 469–471.
  • H. S. Wilf, Some applications of the inequality of arithmetic and geometric means to polynomial equations, Proc. Amer. Math. Soc. 14 (1963), 263–265.