Another approach to characterizations of generalized triangle inequalities in normed spaces

Tamotsu Izumida; Ken-Ichi Mitani; Kichi-Suke Saito

Open Mathematics (2014)

  • Volume: 12, Issue: 11, page 1615-1623
  • ISSN: 2391-5455

Abstract

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In this paper, we consider a generalized triangle inequality of the following type: x 1 + ⋯ + x n p ⩽ x 1 p μ 1 + ⋯ + x 2 p μ n f o r a l l x 1 , ... , x n ∈ X , where (X, ‖·‖) is a normed space, (µ1, ..., µn) ∈ ℝn and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].

How to cite

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Tamotsu Izumida, Ken-Ichi Mitani, and Kichi-Suke Saito. "Another approach to characterizations of generalized triangle inequalities in normed spaces." Open Mathematics 12.11 (2014): 1615-1623. <http://eudml.org/doc/269693>.

@article{TamotsuIzumida2014,
abstract = {In this paper, we consider a generalized triangle inequality of the following type: \[\left\Vert \{x\_1 + \cdots + x\_n \} \right\Vert ^p \leqslant \frac\{\{\left\Vert \{x\_1 \} \right\Vert ^p \}\}\{\{\mu \_1 \}\} + \cdots + \frac\{\{\left\Vert \{x\_2 \} \right\Vert ^p \}\}\{\{\mu \_n \}\}\left( \{for all x\_1 , \ldots ,x\_n \in X\} \right),\] where (X, ‖·‖) is a normed space, (µ1, ..., µn) ∈ ℝn and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].},
author = {Tamotsu Izumida, Ken-Ichi Mitani, Kichi-Suke Saito},
journal = {Open Mathematics},
keywords = {Generalized triangle inequality; Absolute norm; ψ-direct sum; Generalized Hölder inequality; generalized triangle inequality; absolute norm; -direct sum; generalized Hölder inequality},
language = {eng},
number = {11},
pages = {1615-1623},
title = {Another approach to characterizations of generalized triangle inequalities in normed spaces},
url = {http://eudml.org/doc/269693},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Tamotsu Izumida
AU - Ken-Ichi Mitani
AU - Kichi-Suke Saito
TI - Another approach to characterizations of generalized triangle inequalities in normed spaces
JO - Open Mathematics
PY - 2014
VL - 12
IS - 11
SP - 1615
EP - 1623
AB - In this paper, we consider a generalized triangle inequality of the following type: \[\left\Vert {x_1 + \cdots + x_n } \right\Vert ^p \leqslant \frac{{\left\Vert {x_1 } \right\Vert ^p }}{{\mu _1 }} + \cdots + \frac{{\left\Vert {x_2 } \right\Vert ^p }}{{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),\] where (X, ‖·‖) is a normed space, (µ1, ..., µn) ∈ ℝn and p > 0. By using ψ-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which is given by [Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Nonlinear Anal., 2012, 75(2), 735–741].
LA - eng
KW - Generalized triangle inequality; Absolute norm; ψ-direct sum; Generalized Hölder inequality; generalized triangle inequality; absolute norm; -direct sum; generalized Hölder inequality
UR - http://eudml.org/doc/269693
ER -

References

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  1. [1] Ansari A.H., Moslehian M.S., Refinements of reverse triangle inequalities in inner product spaces, J. Inequal. Pure Appl. Math., 2005, 6(3), article 64, 12pp. Zbl1095.46016
  2. [2] Dadipour F., Moslehian M.S., Rassias J.M., Takahasi S.-E., Characterizations of a generalized triangle inequality in normed spaces, Nonlinear Anal., 2012, 75(2), 735–741 http://dx.doi.org/10.1016/j.na.2011.09.004 Zbl1242.46029
  3. [3] Kato M., Saito K.-S., Tamura T., On ψ-direct sums of Banach spaces and convexity, J. Aust. Math. Soc., 2003, 75(3), 413–422 http://dx.doi.org/10.1017/S1446788700008193 Zbl1055.46010
  4. [4] Kato M., Saito K.-S., Tamura T., Sharp triangle inequality and its reverse in Banach spaces, Math. Inequal. Appl., 2007, 10(2), 451–460 Zbl1121.46019
  5. [5] Maligranda L., Some remarks on the triangle inequality for norms, Banach J. Math. Anal., 2008, 2(2), 31–41 Zbl1147.46020
  6. [6] Mitani K.-I., Oshiro S., Saito K.-S., Smoothness of ψ-direct sums of Banach spaces, Math. Ineq. Appl., 2005, 8(1), 147–157 Zbl1084.46012
  7. [7] Mitani K.-I., Saito K.-S., On sharp triangle inequalities in Banach spaces II, J. Inequal. Appl., 2010, Art. ID 323609, 17pp. 
  8. [8] 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.2007.03.036 Zbl1127.46015
  9. [9] Nikolova L., Persson L.-E., Varošanec S., The Beckenbach-Dresher inequality in the -direct sums of spaces and related results, J. Inequal. Appl., 2012, 2012:7, 14pp. http://dx.doi.org/10.1186/1029-242X-2012-7 Zbl1275.26042
  10. [10] Saito K.-S., Kato M., Takahashi Y., Von Neumann-Jordan constant of absolute normalized norms on ℂ2, J. Math. Anal. Appl., 2000, 244(2), 515–532 http://dx.doi.org/10.1006/jmaa.2000.6727 
  11. [11] Saito K.-S., Kato M., Takahashi Y., Absolute norms on ℂn, J. Math. Anal. Appl., 2000, 252(2), 879–905 http://dx.doi.org/10.1006/jmaa.2000.7139 
  12. [12] Takahasi S.-E., Rassias J.M., Saitoh S., Takahashi Y., Refined generalizations of the triangle inequality on Banach spaces, Math. Ineq. Appl., 2010, 13(4), 733–741 

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