## On some generalized triangle inequalities and $$\ell_{\psi}$$-spaces.(English)Zbl 1352.46023

Summary: In this paper, we consider a generalized triangle inequality of the following type: $\| a_1 x_1+\cdots + a_1 x_n \| ^p\leq \| x_1\|^p +\cdots +\| x_n\| ^p \;(x_1,\dots, x_n\in X),$ where $$(X, \| \cdot \|)$$ is a normed space, $$(a_1, \dots, a_n) \in \mathbb C^n$$ and $$p>0$$. By using generalized $$\ell_p$$-spaces, we present a characterization of above inequality for infinite sequences $$\{x_n\}_{n=1}^{\infty} \subset X$$.

### MSC:

 46B99 Normed linear spaces and Banach spaces; Banach lattices
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### References:

 [1] S. Abramovich, S. Ivelić and J. Pečarić, Extension of the Euler-Lagrange identity by superquadratic power functions , International J. Pure and Applied Math. 74 (2012), 209-220. · Zbl 1244.26032 [2] F. Dadipour, M.S. Moslehian, J.M. Rassias and S.-E. Takahasi, Characterizations of a generalized triangle inequality in normed spaces , Nonlinear Analysis 75 (2012), 735-741. · Zbl 1242.46029 [3] T. Izumida, K.-I. Mitani and K.-S. Saito, Another approach to characterizations of generalized triangle inequalities in normed spaces , Cent. Eur. J. Math. 12 (2014), 1615-1623. · Zbl 1311.46014 [4] M. Kato, K.-S. Saito and T. Tamura, On $$\psi$$-direct sums of Banach spaces and convexity , J. Aust. Math. Soc. 75 (2003), 413-422. · Zbl 1055.46010 [5] K.-I. Mitani, S. Oshiro and K.-S. Saito, Smoothness of $$\psi$$-direct sums of Banach spaces , Math. Inequal. Appl. 8 (2005), 147-157. · Zbl 1084.46012 [6] K.-I. Mitani and K.-S. Saito, On generalized $$\ell_p$$-spaces , Hiroshima Math. J. 37 (2007), 1-12. [7] L. Nikolova, L.-E. Persson and S. Varošanec, The Beckenbach-Dresher inequality in the $$\psi$$-direct sums of spaces and related results , J. Inequal. Appl. 2012 , 2012:7, 14pp. · Zbl 1275.26042 [8] K.-S. Saito, M. Kato and Y. Takahashi, Von Neumann-Jordan constant of absolute normalized norms on $$\mathbb C^2$$ , J. Math. Anal. Appl. 244 (2000), 515-532. · Zbl 0961.46008 [9] K.-S. Saito, M. Kato and Y. Takahashi, Absolute norms on $$\mathbb C^n$$ , J. Math. Anal. Appl. 252 (2000), 879-905. · Zbl 0999.46008 [10] S.-E. Takahasi, J.M. Rassias, S. Saitoh and Y. Takahashi, Refined generalizations of the triangle inequality on Banach spaces , Math. Inequal. Appl. 13 (2010), 733-741. [11] T. Zachariades, On $$\ell_{\psi}$$ spaces and $$\psi$$-direct sums of Banach space , Rocky Mountain J. Math. 41 (2011), 971-997. · Zbl 1235.46025
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