Annals of Functional Analysis

Matrix inequalities for the difference between arithmetic mean and harmonic mean

Wenshi Liao and Junliang Wu

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

Motivated by the refinements and reverses of arithmetic-geometric mean and arithmetic-harmonic mean inequalities for scalars and matrices, in this article, we generalize the scalar and matrix inequalities for the difference between arithmetic mean and harmonic mean. In addition, relevant inequalities for the Hilbert-Schmidt norm and determinant are established.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 3 (2015), 191-202.

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.afa/1429286042

Digital Object Identifier
doi:10.15352/afa/06-3-16

Mathematical Reviews number (MathSciNet)
MR3336915

Zentralblatt MATH identifier
06441318

Subjects
Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 26E60: Means [See also 47A64] 47A63: Operator inequalities

Keywords
Arithmetic mean harmonic mean matrix inequality positive definite matrix

Citation

Liao, Wenshi; Wu, Junliang. Matrix inequalities for the difference between arithmetic mean and harmonic mean. Ann. Funct. Anal. 6 (2015), no. 3, 191--202. doi:10.15352/afa/06-3-16. https://projecteuclid.org/euclid.afa/1429286042


Export citation

References

  • H. Alzer, C.M. da Fonseca and A. Kovačec, Young-type inequalities and their matrix analogues, Linear Multilinear Algebra 63 (2015), no. 3, 622–635.
  • T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 ( 1979), 203–241.
  • T. Ando, On the arithmetic-geometric-harmonic-Mean inequalities for positive definite Matrices, Linear Algebra Appl. 52/53 (1983), 31–37.
  • R. Bhatia, Matrix Analysis, Springer, New York, 1997.
  • S. Furuichi, On refined Young inequalities and reverse inequalities, J. Math. Inequal. 5 (2011), no. 1, 21–31.
  • O. Hirzallah and F. Kittaneh, Matrix Young inequalities for the Hilbert-Schmidt norm, Linear Algebra Appl. 308 (2000), 77–84.
  • O. Hirzallah, F. Kittaneh, M. Krnić, N. Lovričević and J. Pečarić, Refinements and reverses of means inequalities for Hilbert space operators, Banach J. Math. Anal. 7 (2013), no. 2, 15–29.
  • R.A. Horn and C.R. Johnson, Matrix analysis, 2nd ed, Cambridge University Press, New York, 2013.
  • F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra. 59 (2011), no. 9, 1031–1037.
  • M. Krnić, N. Lovričević and J. Pečarić, Jensen's operator and applications to mean inequalities for operators in Hilbert space, Bull. Malays. Math. Sci. Soc. 35 (2012), no. 1, 1–14.
  • A.W. Roberts and D.E. Varberg, Convex Functions, Academic Press, New York, 1973.
  • M. Sagae and K. Tanabe, Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices, Linear Multilinear Algebra. 37 (1994), 279–282.
  • A. Salemi and A.S. Hosseini, On reversing of the modified Young inequality, Ann. Funct. Anal. 5 (2014), no. 1, 70–76.
  • H.L. Zuo, G.H. Shi and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5 (2011), no. 4, 551–556.