Another approach to characterizations of generalized triangle inequalities in normed spaces

• Autorzy: Tamotsu Izumida , Ken-Ichi Mitani , Kichi-Suke Saito
• Języki publikacji: EN

Abstrakty

EN

In this paper, we consider a generalized triangle inequality of the following type: $$\left\| {x_1 + \cdots + x_n } \right\|^p \leqslant \frac{{\left\| {x_1 } \right\|^p }} {{\mu _1 }} + \cdots + \frac{{\left\| {x_2 } \right\|^p }} {{\mu _n }}\left( {for all x_1 , \ldots ,x_n \in X} \right),$$ 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].

Słowa kluczowe

EN

Generalized triangle inequality; Absolute norm; ψ-direct sum; Generalized Hölder inequality

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B25: Classical Banach spaces in the general theory

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 11
• Strony: 1615-1623

Daty

wydano

2014-11-01

online

2014-06-29

Twórcy

autor

Tamotsu Izumida

• Niigata University

autor

Ken-Ichi Mitani

• Okayama Prefectural University

autor

Kichi-Suke Saito

• Niigata University

Bibliografia

• [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.
• [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
• [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
• [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
• [5] Maligranda L., Some remarks on the triangle inequality for norms, Banach J. Math. Anal., 2008, 2(2), 31–41
• [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
• [7] Mitani K.-I., Saito K.-S., On sharp triangle inequalities in Banach spaces II, J. Inequal. Appl., 2010, Art. ID 323609, 17pp.
• [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
• [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
• [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] 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] 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

Identyfikatory

DOI

10.2478/s11533-014-0432-z