Characterization of intermediate values of the triangle inequality II

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

Abstrakty

EN

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.

Słowa kluczowe

EN

Triangle inequalities; Strictly convex Banach spaces; Norm inequality

Kategorie tematyczne

• 46B99: None of the above, but in this section
• 46B20: Geometry and structure of normed linear spaces
• 26D20: Other analytical inequalities

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 5
• Strony: 778-786

Daty

wydano

2014-05-01

online

2014-02-15

Twórcy

autor

Hiroki Sano

• Shizuoka University

autor

Tamotsu Izumida

• Niigata University

autor

Ken-Ichi Mitani

• Okayama Prefectural University

autor

Tomoyoshi Ohwada

• Shizuoka University

autor

Kichi-Suke Saito

• Niigata University

Bibliografia

• [1] Abramovich Y.A., Aliprantis C.D., Problems in Operator Theory, Grad. Stud. in Math., 51, American Mathematical Society, Providence, 2002
• [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.2005.2883
• [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.2011.09.004
• [4] Dragomir S.S., Reverses of the triangle inequality in Banach spaces, JIPAM. J. Inequal. Pure Appl. Math., 2005, 6(5), #129
• [5] Dragomir S.S., Generalizations of the Pečarić-Rajić inequality in normed linear spaces, Math. Inequal. Appl., 2009, 12(1), 53–65
• [6] Fujii M., Kato M., Saito K.-S., Tamura T., Sharp mean triangle inequality, Math. Inequal. Appl., 2010, 13(4), 743–752
• [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.2008.01.088
• [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
• [9] Maligranda L., Some remarks on the triangle inequality for norms, Banach J. Math. Anal., 2008, 2(2), 31–41
• [10] Martirosyan M.S., Samarchyan S.V., Inversion of the triangle inequality in ℝn, J. Contemp. Math. Anal., 2003, 38(4), 56–61
• [11] Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035
• [12] Mitani K.-I., Saito K.-S., On sharp triangle inequalities in Banach spaces II, J. Inequal. Appl., 2010, #323609
• [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.2007.03.036
• [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/s11533-013-0238-4
• [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
• [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
• [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
• [18] Saitoh S., Generalizations of the triangle inequality, JIPAM. J. Inequal. Pure Appl. Math., 2003, 43), #62
• [19] Zhang L., Ohwada T., Chō M., Reverses of the triangle inequality in inner product spaces, Math. Inequal. Appl. (in press)

Identyfikatory

DOI

10.2478/s11533-013-0369-7