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

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 12, Issue 5

Issues

Volume 13 (2015)

Characterization of intermediate values of the triangle inequality II

Hiroki Sano / Tamotsu Izumida
  • Department of Mathematical Sciences, Graduate School of Science and Technology, Niigata University, Niigata, 950-2181, Japan
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Ken-Ichi Mitani
  • Department of Information and Communication Engineering, Faculty of Computer Science and System Engineering, Okayama Prefectural University, Okayama, 719-1197, Japan
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Tomoyoshi Ohwada / Kichi-Suke Saito
Published Online: 2014-02-15 | DOI: https://doi.org/10.2478/s11533-013-0369-7

Abstract

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C n that satisfy 0 ≤ C n ≤ Σj=1n ‖x j‖ − ‖Σj=1n x j‖, x 1,...,x n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.

Keywords: Triangle inequalities; Strictly convex Banach spaces; Norm inequality

MSC: 46B20; 46B99; 26D20

  • [1] Abramovich Y.A., Aliprantis C.D., Problems in Operator Theory, Grad. Stud. in Math., 51, American Mathematical Society, Providence, 2002 Google Scholar

  • [2] Ansari A.H., Moslehian M.S., More on reverse triangle inequality in inner product spaces, Int. J. Math. Math. Sci., 2005, 18, 2883–2893 http://dx.doi.org/10.1155/IJMMS.2005.2883CrossrefGoogle Scholar

  • [3] Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Characterization of a generalized triangle inequality in normed spaces, Nonlinear Anal., 2012, 75(2), 735–741 http://dx.doi.org/10.1016/j.na.2011.09.004CrossrefGoogle Scholar

  • [4] Dragomir S.S., Reverses of the triangle inequality in Banach spaces, JIPAM. J. Inequal. Pure Appl. Math., 2005, 6(5), #129 Google Scholar

  • [5] Dragomir S.S., Generalizations of the Pečarić-Rajić inequality in normed linear spaces, Math. Inequal. Appl., 2009, 12(1), 53–65 Google Scholar

  • [6] Fujii M., Kato M., Saito K.-S., Tamura T., Sharp mean triangle inequality, Math. Inequal. Appl., 2010, 13(4), 743–752 Google Scholar

  • [7] Hsu C.-Y., Shaw S.-Y., Wong H.-J., Refinements of generalized triangle inequalities, J. Math. Anal. Appl., 2008, 344(1), 17–31 http://dx.doi.org/10.1016/j.jmaa.2008.01.088CrossrefGoogle Scholar

  • [8] Kato M., Saito K.-S., Tamura T., Sharp triangle inequality and its reverse in Banach spaces, Math. Inequal. Appl., 2007, 10(2), 451–460 Google Scholar

  • [9] Maligranda L., Some remarks on the triangle inequality for norms, Banach J. Math. Anal., 2008, 2(2), 31–41 Google Scholar

  • [10] Martirosyan M.S., Samarchyan S.V., Inversion of the triangle inequality in ℝn, J. Contemp. Math. Anal., 2003, 38(4), 56–61 Google Scholar

  • [11] Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035 Web of ScienceGoogle Scholar

  • [12] Mitani K.-I., Saito K.-S., On sharp triangle inequalities in Banach spaces II, J. Inequal. Appl., 2010, #323609 Google Scholar

  • [13] Mitani K.-I., Saito K.-S., Kato M., Tamura T., On sharp triangle inequalities in Banach spaces, J. Math. Anal. Appl., 2007, 336(2), 1178–1186 http://dx.doi.org/10.1016/j.jmaa.2007.03.036CrossrefGoogle Scholar

  • [14] Mizuguchi H., Saito K.-S., Tanaka R., On the calculation of the Dunkl-Williams constant of normed linear spaces, Cent. Eur. J. Math., 2013, 11(7), 1212–1227 http://dx.doi.org/10.2478/s11533-013-0238-4CrossrefWeb of ScienceGoogle Scholar

  • [15] Moslehian M.S., Dadipour F., Rajic R., Maric A., A glimpse at the Dunkl-Williams inequality, Banach J. Math. Anal., 2011, 5(2), 138–151 Google Scholar

  • [16] Ohwada T., On a continuous mapping and sharp triangle inequalities, In: Inequalities and Applications 2010, International Series of Numerical Mathematics, 161, Springer, Basel, 2011, 125–136 Google Scholar

  • [17] Saito K.-S., Mitani K.-I., On sharp triangle inequalities in Banach spaces and their applications, In: Banach and Function Spaces III, Yokohama Publications, Yokohama, 2011, 295–304 Google Scholar

  • [18] Saitoh S., Generalizations of the triangle inequality, JIPAM. J. Inequal. Pure Appl. Math., 2003, 43), #62 Google Scholar

  • [19] Zhang L., Ohwada T., Chō M., Reverses of the triangle inequality in inner product spaces, Math. Inequal. Appl. (in press) Web of ScienceGoogle Scholar

About the article

Published Online: 2014-02-15

Published in Print: 2014-05-01


Citation Information: Open Mathematics, Volume 12, Issue 5, Pages 778–786, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0369-7.

Export Citation

© 2014 Versita Warsaw. 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