Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Tbilisi Mathematical Journal

Editor-in-Chief: Inassaridze, Hvedri

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.14

Open Access
Online
ISSN
1512-0139
See all formats and pricing
More options …

Vector norm inequalities for power series of operators in Hilbert spaces

W.-S. Cheung
  • Corresponding author
  • Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong, China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ S.S. Dragomir
  • Corresponding author
  • Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
  • School of Computational & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-12-30 | DOI: https://doi.org/10.2478/tmj-2014-0013

Abstract

In this paper, vector norm inequalities that provides upper bounds for the Lipschitz quantity ║f (T) x - f (V ) x ║ for power series f(z) = ∑n=0 anzn; bounded linear operators T; V on the Hilbert space H and vectors x ∈ H are established. Applications in relation to Hermite- Hadamard type inequalities and examples for elementary functions of interest are given as well.

Keywords: Bounded linear operators; Hilbert spaces; Functions of operators; Power series; Hermite-Hadamard type inequal- ities

References

  • [1] H. Araki and S. Yamagami, An inequality for Hilbert-Schmidt norm, Commun. Math. Phys. 81 (1981), 89-96. Google Scholar

  • [2] R. Bhatia, First and second order perturbation bounds for the operator absolute value, Linear Algebra Appl. 208/209 (1994), 367-376. Google Scholar

  • [3] R. Bhatia, Perturbation bounds for the operator absolute value. Linear Algebra Appl. 226/228 (1995), 639-645. Google Scholar

  • [4] R. Bhatia, D. Singh and K. B. Sinha, Differentiation of operator functions and perturbation bounds. Comm. Math. Phys. 191 (1998), no. 3, 603-611. Google Scholar

  • [5] R. Bhatia, Matrix Analysis, Springer Verlag, 1997. Google Scholar

  • [6] L. Ciurdariu, A note concerning several Hermite-Hadamard inequalities for different types of convex functions. Int. J. Math. Anal. 6 (2012), no. 33-36, 1623-1639. Google Scholar

  • [7] S. S. Dragomir, Y. J. Cho and S. S. Kim, Inequalities of Hadamard's type for Lipschitzian mappings and their applications. J. Math. Anal. Appl. 245 (2000), no. 2, 489-501. Google Scholar

  • [8] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite- Hadamard Inequalities and Applications, RGMIA Monographs, 2000. [Online http://rgmia.org/monographs/hermite hadamard.html]. Google Scholar

  • [9] S. S. Dragomir, Inequalities of Lipschitz type for power series of operators in Hilbert spaces, Commun. Math. Anal. 16 (2014), Number 1, 102 - 122. Google Scholar

  • [10] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972. Google Scholar

  • [11] Yu. B. Farforovskaya, Estimates of the closeness of spectral decompositions of self-adjoint op- erators in the Kantorovich-Rubinshtein metric (in Russian), Vesln. Leningrad. Gos. Univ. Ser. Mat. Mekh. Astronom. 4 (1967), 155-156. Google Scholar

  • [12] Yu. B. Farforovskaya, An estimate of the norm ||f (B) - f (A)|| for self-adjoint operators A and B (in Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. 56 (1976), 143-162 . Google Scholar

  • [13] Yu. B. Farforovskaya and L. Nikolskaya, Modulus of continuity of operator functions. Algebra i Analiz 20 (2008), no. 3, 224-242; translation in St. Petersburg Math. J. 20 (2009), no. 3, 493-506. Google Scholar

  • [14] Y. Feng and W. Zhao, Renement of Hermite-Hadamard inequality. Far East J. Math. Sci. 68 (2012), no. 2, 245-250. Google Scholar

  • [15] X. Gao, A note on the Hermite-Hadamard inequality. J. Math. Inequal. 4 (2010), no. 4, 587- 591. CrossrefGoogle Scholar

  • [16] S.-R. Hwang, K.-L. Tseng and K.-C. Hsu, Hermite-Hadamard type and Fejer type inequalities for general weights (I). J. Inequal. Appl. 2013, 2013:170. Google Scholar

  • [17] T. Kato, Continuity of the map S ! jSj for linear operators, Proc. Japan Acad. 49 (1973), 143-162. Google Scholar

  • [18] U. S. Krmac and R. Dikici, On some Hermite-Hadamard type inequalities for twice differen- tiable mappings and applications. Tamkang J. Math. 44 (2013), no. 1, 41-51. Google Scholar

  • [19] J. Mikusinski, The Bochner Integral, Birkhauser Verlag, 1978. Google Scholar

  • [20] M. Muddassar, M. I. Bhatti and M. Iqbal, Some new s-Hermite-Hadamard type inequalities for differentiable functions and their applications. Proc. Pakistan Acad. Sci. 49 (2012), no. 1, 9-17. Google Scholar

  • [21] M. Matic and J. Pecaric, Note on inequalities of Hadamard's type for Lipschitzian mappings. Tamkang J. Math. 32 (2001), no. 2, 127-130. Google Scholar

  • [22] W. Rudin, Functional Analysis, McGraw Hill, 1973. Google Scholar

  • [23] M. Z. Sarikaya, On new Hermite Hadamard Fejer type integral inequalities. Stud. Univ. Babes- Bolyai Math. 57 (2012), no. 3, 377-386. Google Scholar

  • [24] S. Wasowicz and A. Witkowski, On some inequality of Hermite-Hadamard type. Opuscula Math. 32 (2012), no. 3, 591-600. Google Scholar

  • [25] B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means. J. Funct. Spaces Appl. 2012, Art. ID 980438, 14 pp. Web of ScienceGoogle Scholar

  • [26] G. Zabandan, A. Bodaghi and A. Klcman, The Hermite-Hadamard inequality for r-convex functions. J. Inequal. Appl. 2012, 2012:215, 8 pp. CrossrefGoogle Scholar

  • [27] C.-J. Zhao, W.-S. Cheung and X.-Y. Li, On the Hermite-Hadamard type inequalities. J. Inequal. Appl. 2013, 2013:228. Google Scholar

About the article

Published Online: 2014-12-30


Citation Information: Tbilisi Mathematical Journal, Volume 7, Issue 2, ISSN (Online) 1512-0139, DOI: https://doi.org/10.2478/tmj-2014-0013.

Export Citation

© 2014 . This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in