Izumida, Tamotsu; Mitani, Ken-Ichi; Saito, Kichi-Suke Another approach to characterizations of generalized triangle inequalities in normed spaces. (English) Zbl 1311.46014 Cent. Eur. J. Math. 12, No. 11, 1615-1623 (2014). Summary: In this paper, we consider a generalized triangle inequality of the following type: \[ \| x_1+\cdots+x_n\|^p\leq\frac{\| x_1\|^p}{\mu_1}+\cdots+\frac{\| x_2\|^p}{\mu_n} \quad \text{(for all }x_1,\dots,x_n\in X), \] where \((X,\|\cdot\|)\) is a normed space, \((\mu_1,\dots,\mu_n)\in\mathbb R^n\) and \(p>0\). By using \(\psi\)-direct sums of Banach spaces, we present another approach to characterizations of the above inequality which has been given by F. Dadipour et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 2, 735–741 (2012; Zbl 1242.46029)]. Cited in 4 Documents MSC: 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory Keywords:generalized triangle inequality; absolute norm; \(\psi\)-direct sum; generalized Hölder inequality Citations:Zbl 1242.46029 PDFBibTeX XMLCite \textit{T. Izumida} et al., Cent. Eur. J. Math. 12, No. 11, 1615--1623 (2014; Zbl 1311.46014) Full Text: DOI References: [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.; · Zbl 1095.46016 [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; · Zbl 1242.46029 [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; · Zbl 1055.46010 [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; · Zbl 1121.46019 [5] Maligranda L., Some remarks on the triangle inequality for norms, Banach J. Math. Anal., 2008, 2(2), 31-41; · Zbl 1147.46020 [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; · Zbl 1084.46012 [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; · Zbl 1127.46015 [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; · Zbl 1275.26042 [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; · Zbl 0961.46008 [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; · Zbl 0999.46008 [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; · Zbl 1205.26034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.