Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 514184, 5 pages
http://dx.doi.org/10.5402/2011/514184
Research Article

Another Aspect of Triangle Inequality

1Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan
2Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
3Department of Mathematics and Information Science, Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
4Department of Systems Engineering, Okayama Prefectural University, Soja, Okayama 719-1197, Japan

Received 18 February 2011; Accepted 14 March 2011

Academic Editors: Y.Β Dai and B.Β Djafari-Rouhani

Copyright Β© 2011 Kichi-Suke Saito et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the notion of πœ“-norm by considering the fact that an absolute normalized norm on β„‚2 corresponds to a continuous convex function πœ“ on the unit interval [0,1] with some conditions. This is a generalization of the notion of π‘ž-norm introduced by Belbachir et al. (2006). Then we show that a πœ“-norm is a norm in the usual sense.

1. Introduction

The triangle inequality is one of the most fundamental inequalities in analysis and has been studied by several authors. For example, Kato et al. in [1] showed a sharpened triangle inequality and its reverse one with 𝑛 elements in a Banach space (see also [2–4]). Here we consider another aspect of the classical triangle inequality β€–π‘₯+𝑦‖≀‖π‘₯β€–+‖𝑦‖. For a Hilbert space 𝐻, we recall the parallelogram law β€–π‘₯+𝑦‖2+β€–π‘₯βˆ’π‘¦β€–2ξ€·=2β€–π‘₯β€–2+‖𝑦‖2ξ€Έ(π‘₯,π‘¦βˆˆπ»).(1.1) This implies that the parallelogram inequality β€–π‘₯+𝑦‖2≀2β€–π‘₯β€–2+‖𝑦‖2ξ€Έ(π‘₯,π‘¦βˆˆπ»)(1.2) holds. Saitoh in [5] noted the inequality (1.2) may be more suitable than the classical triangle inequality and used the inequality (1.2) to the setting of a natural sum Hilbert space for two arbitrary Hilbert spaces. Motivated by this, Belbachir et al. [6] introduced the notion of π‘ž-norm (1β‰€π‘ž<∞) in a vector space 𝑋 over 𝕂(=ℝ or β„‚), where the definition of π‘ž-norm is a mapping β€–β‹…β€– from 𝑋 into ℝ+(={π‘Žβˆˆβ„βˆΆπ‘Žβ‰₯0}) satisfying the following conditions: (i)β€–π‘₯β€–=0⇔π‘₯=0, (ii)‖𝛼π‘₯β€–=|𝛼|β€–π‘₯β€–(π‘₯βˆˆπ‘‹,π›Όβˆˆπ•‚), (iii)β€–π‘₯+π‘¦β€–π‘žβ‰€2π‘žβˆ’1(β€–π‘₯β€–π‘ž+β€–π‘¦β€–π‘ž)(π‘₯,π‘¦βˆˆπ‘‹).

We easily show that every norm is a π‘ž-norm. Conversely, they proved that for all π‘ž with 1β‰€π‘ž<∞, every π‘ž-norm is a norm in the usual sense.

In this paper, we generalize the notion of π‘ž-norm, that is, we introduce the notion of πœ“-norm by considering the fact that an absolute normalized norm on ℝ2 corresponds to a continuous convex function πœ“ on the unit interval [0,1] with some conditions (cf. [7]). We show that a πœ“-norm is a norm in the usual sense.

We recall some properties of absolute normalized norms on β„‚2. A norm β€–β‹…β€– on β„‚2 is called absolute if β€–(π‘₯,𝑦)β€–=β€–(|π‘₯|,|𝑦|)β€– for all (π‘₯,𝑦)βˆˆβ„‚2 and normalized if β€–(1,0)β€–=β€–(0,1)β€–=1. The ℓ𝑝-norms ‖⋅‖𝑝 are such examples: β€–(π‘₯,𝑦)‖𝑝=ξƒ―ξ€·|π‘₯|𝑝+||𝑦||𝑝1/𝑝||𝑦||ξ€Ύif1≀𝑝<∞,max|π‘₯|,if𝑝=∞.(1.3) Let 𝐴𝑁2 be the family of all absolute normalized norms on β„‚2. It is well known that the set 𝐴𝑁2 is a one-to-one correspondence with the set Ξ¨2 of all continuous convex functions πœ“ on the unit interval [0,1] satisfying max{1βˆ’π‘‘,𝑑}β‰€πœ“(𝑑)≀1 for 𝑑 with 0≀𝑑≀1 (see [7, 8]). The correspondence is given by the equation πœ“(𝑑)=β€–(1βˆ’π‘‘,𝑑)β€–πœ“. Indeed, for all πœ“ in Ξ¨2, we define the norm β€–β‹…β€–πœ“ as β€–(π‘₯,𝑦)β€–πœ“=⎧βŽͺ⎨βŽͺβŽ©ξ€·||𝑦||ξ€Έπœ“ξ‚΅||𝑦|||π‘₯|+||𝑦||ξ‚Ά|π‘₯|+if(π‘₯,𝑦)β‰ (0,0),0if(π‘₯,𝑦)=(0,0).(1.4) Then β€–β‹…β€–πœ“βˆˆπ΄π‘2 and satisfies πœ“(𝑑)=β€–(1βˆ’π‘‘,𝑑)β€–πœ“. The functions which correspond to the ℓ𝑝-norms ‖⋅‖𝑝 on β„‚2 are πœ“π‘(𝑑)={(1βˆ’π‘‘)𝑝+𝑑𝑝}1/𝑝 if 1≀𝑝<∞ and πœ“βˆž(𝑑)=max{1βˆ’π‘‘,𝑑} if 𝑝=∞.

2. πœ“-Norm

Definition 2.1. Let 𝑋 be a vector space and πœ“βˆˆΞ¨2. Then a mapping β€–β‹…β€–βˆΆπ‘‹β†’β„+ is called πœ“-norm on 𝑋 if it satisfies the following conditions: (i)β€–π‘₯β€–=0⇔π‘₯=0,(ii)‖𝛼π‘₯β€–=|𝛼|β€–π‘₯β€–(π‘₯βˆˆπ‘‹,π›Όβˆˆπ•‚),(iii)β€–π‘₯+𝑦‖≀(1/min0≀𝑑≀1πœ“(𝑑))β€–(β€–π‘₯β€–,‖𝑦‖)β€–πœ“(π‘₯,π‘¦βˆˆπ‘‹).

Note that for all π‘ž with 1β‰€π‘ž<∞, any πœ“π‘ž-norm β€–β‹…β€– is just a π‘ž-norm. Indeed, since the function πœ“π‘ž takes the minimum at 𝑑=1/2 and πœ“π‘žξ‚€12=ξ‚΅ξ‚€12ξ‚π‘ž+ξ‚€12ξ‚π‘žξ‚Ά1/π‘ž=21/π‘žβˆ’1,(2.1) the condition (iii) of Definition 2.1 implies 1β€–π‘₯+π‘¦β€–β‰€πœ“π‘ž(1/2)β€–(β€–π‘₯β€–,‖𝑦‖)β€–πœ“π‘ž=21βˆ’1/π‘žξ€·β€–π‘₯β€–π‘ž+β€–π‘¦β€–π‘žξ€Έ1/π‘ž.(2.2) Thus we have β€–π‘₯+π‘¦β€–π‘žβ‰€2π‘žβˆ’1(β€–π‘₯β€–π‘ž+β€–π‘¦β€–π‘ž) and so β€–β‹…β€– becomes a π‘ž-norm.

If πœ“=πœ“1, then the condition (iii) of Definition 2.1 is just a triangle inequality. Thus we suppose that πœ“β‰ πœ“1.

Proposition 2.2. Let 𝑋 be a vector space and πœ“βˆˆΞ¨2 with πœ“β‰ πœ“1. Then every norm on 𝑋 in the usual sense is a πœ“-norm.

To do this, we need the following lemma given in [7].

Lemma 2.3 (see [7]). Let πœ“,πœ‘βˆˆΞ¨2 and πœ‘β‰₯πœ“. Put 𝑀=max0≀𝑑≀1πœ‘(𝑑)πœ“(𝑑).(2.3) Then β€–β‹…β€–πœ“β‰€β€–β‹…β€–πœ‘β‰€π‘€β€–β‹…β€–πœ“.(2.4)

Proof of Proposition 2.2. Let β€–β‹…β€– be a norm on 𝑋 and π‘₯,π‘¦βˆˆπ‘‹. Since πœ“β‰€πœ“1, we have by Lemma 2.3, β€–π‘₯+𝑦‖≀‖π‘₯β€–+‖𝑦‖=β€–(β€–π‘₯β€–,‖𝑦‖)β€–1≀max0≀𝑑≀11πœ“β€–(𝑑)(β€–π‘₯β€–,‖𝑦‖)β€–πœ“=1min0≀𝑑≀1πœ“(𝑑)β€–(β€–π‘₯β€–,‖𝑦‖)β€–πœ“.(2.5) Thus β€–β‹…β€– is a πœ“-norm on 𝑋.

We will show that every πœ“-norm is a norm in the usual sense. To do this, we need the following lemma given in [6].

Lemma 2.4 (see [6]). Let 𝑋 be a vector space. Let β€–β‹…β€–βˆΆπ‘‹β†’β„+ be a mapping satisfying the conditions (i) and (ii) in Definition 2.1. Then β€–β‹…β€– is a norm if and only if the set 𝐡𝑋={π‘₯βˆˆπ‘‹βˆΆβ€–π‘₯‖≀1} is convex.

Proof. Suppose that 𝐡𝑋 is convex. For every π‘₯,π‘¦βˆˆπ‘‹ such that π‘₯β‰ 0,𝑦≠0, we have β€–β€–β€–π‘₯+𝑦‖π‘₯β€–+‖𝑦‖‖‖‖=β€–β€–β€–β€–π‘₯β€–+‖𝑦‖‖π‘₯β€–π‘₯β€–π‘₯β€–+‖𝑦‖+β€–π‘₯‖‖𝑦‖𝑦‖π‘₯β€–+‖𝑦‖‖‖‖‖𝑦‖≀1.(2.6) This completes the proof.

Since every πœ“1-norm is just a usual norm, we suppose that πœ“βˆˆΞ¨2 with πœ“β‰ πœ“1. Put 𝑑0 with 0<𝑑0<1 such that min0≀𝑑≀1πœ“(𝑑)=πœ“(𝑑0). Then we have the following lemma.

Lemma 2.5. Let β€–β‹…β€– be a πœ“-norm on 𝑋. Then, for every π‘₯,π‘¦βˆˆπ΅π‘‹ we have (1βˆ’π‘‘0)π‘₯+𝑑0π‘¦βˆˆπ΅π‘‹.

Proof. Let π‘₯,π‘¦βˆˆπ΅π‘‹. We may assume that π‘₯≠𝑦 and π‘₯,𝑦≠0. From the definition of a πœ“-norm and Lemma  1 in [8], we have β€–β€–ξ€·1βˆ’π‘‘0ξ€Έπ‘₯+𝑑0𝑦‖‖≀1πœ“ξ€·π‘‘0ξ€Έβ€–β€–ξ€·ξ€·1βˆ’π‘‘0ξ€Έβ€–π‘₯β€–,𝑑0ξ€Έβ€–β€–β€–π‘¦β€–πœ“β‰€1πœ“ξ€·π‘‘0ξ€Έβ€–β€–(1βˆ’π‘‘0,𝑑0)β€–β€–πœ“=1,(2.7) which implies (1βˆ’π‘‘0)π‘₯+𝑑0π‘¦βˆˆπ΅π‘‹.

Here we define the set 𝐴𝑛 for all 𝑛=1,2,…, by 𝐴0={0,1},𝐴𝑛=ξ€½ξ€·1βˆ’π‘‘0ξ€Έπ‘Ž+𝑑0π‘βˆΆπ‘Ž,π‘βˆˆπ΄π‘›βˆ’1ξ€Ύ(𝑛=1,2,…).(2.8) Put ⋃𝐴=βˆžπ‘›=0𝐴𝑛. It is clear that 𝐴=[0,1]. We also define a function 𝑓 by 𝑓(π‘₯,𝑦,𝑑)=(1βˆ’π‘‘)π‘₯+𝑑𝑦 for all π‘₯,π‘¦βˆˆπ΅π‘‹ and all π‘‘βˆˆ[0,1].

Lemma 2.6. For every π‘₯,π‘¦βˆˆπ΅π‘‹, we have 𝑓(π‘₯,𝑦,𝑑)βˆˆπ΅π‘‹ for all π‘‘βˆˆπ΄.

Proof. Let π‘₯,π‘¦βˆˆπ΅π‘‹. It is clear that 𝑓(π‘₯,𝑦,𝑑)βˆˆπ΅π‘‹ for all π‘‘βˆˆπ΄0. We suppose that 𝑓(π‘₯,𝑦,𝑑)βˆˆπ΅π‘‹ for all π‘‘βˆˆπ΄π‘›βˆ’1. Then, for all π‘‘βˆˆπ΄π‘›, there exist π‘Ž,π‘βˆˆπ΄π‘›βˆ’1 such that 𝑑=(1βˆ’π‘‘0)π‘Ž+𝑑0𝑏. Hence =𝑓(π‘₯,𝑦,𝑑)=(1βˆ’π‘‘)π‘₯+𝑑𝑦1βˆ’ξ€·ξ€·1βˆ’π‘‘0ξ€Έπ‘Ž+𝑑0𝑏π‘₯+ξ€·ξ€·1βˆ’π‘‘0ξ€Έπ‘Ž+𝑑0𝑏𝑦=ξ€·1βˆ’π‘‘0ξ€Έ((1βˆ’π‘Ž)π‘₯+π‘Žπ‘¦)+𝑑0=ξ€·((1βˆ’π‘)π‘₯+𝑏𝑦)1βˆ’π‘‘0𝑓(π‘₯,𝑦,π‘Ž)+𝑑0𝑓(π‘₯,𝑦,𝑏).(2.9) Since 𝑓(π‘₯,𝑦,π‘Ž) and 𝑓(π‘₯,𝑦,𝑏) are in 𝐡𝑋, we have from Lemma 2.5, 𝑓(π‘₯,𝑦,𝑑)βˆˆπ΅π‘‹ for all π‘‘βˆˆπ΄π‘›. Thus 𝑓(π‘₯,𝑦,𝑑)βˆˆπ΅π‘‹ for all π‘‘βˆˆπ΄.

Theorem 2.7. Let 𝑋 be a vector space and πœ“βˆˆΞ¨2 with πœ“β‰ πœ“1. Then every πœ“-norm on 𝑋 is a norm in the usual sense.

Proof. Let π‘₯,π‘¦βˆˆπ΅π‘‹ and πœ† with 0<πœ†<1. Let 𝑧=(1βˆ’πœ†)π‘₯+πœ†π‘¦. Take a strictly decreasing sequence {π‘Ÿπ‘›} in 𝐴 such that π‘Ÿπ‘›β†˜πœ†. For each 𝑛, we define 𝛽𝑛=(1βˆ’π‘Ÿπ‘›)/(1βˆ’πœ†). Then 0<𝛽𝑛<1 and 𝛽𝑛↗1. Since 0<πœ†π›½π‘›/π‘Ÿπ‘›<1, we have (πœ†π›½π‘›/π‘Ÿπ‘›)π‘¦βˆˆπ΅π‘‹. By Lemma 2.6, 𝛽𝑛𝑧=(1βˆ’πœ†)𝛽𝑛π‘₯+πœ†π›½π‘›π‘¦=ξ€·1βˆ’π‘Ÿπ‘›ξ€Έπ‘₯+π‘Ÿπ‘›πœ†π›½π‘›π‘Ÿπ‘›π‘¦ξ‚΅=𝑓π‘₯,πœ†π›½π‘›π‘Ÿπ‘›π‘¦,π‘Ÿπ‘›ξ‚Άβˆˆπ΅π‘‹.(2.10) Since 𝛽𝑛‖𝑧‖=‖𝛽𝑛𝑧‖≀1, we get π‘§βˆˆπ΅π‘‹. Thus 𝐡𝑋 is convex. By Lemma 2.4, β€–β‹…β€– becomes a norm. This completes the proof.

Acknowledgments

Kichi-Suke Saito was supported in part by Grants-in-Aid for Scientific Research (No. 20540158), Japan Society for the Promotion of Science. Runling An was supported by Program for Top Young Academic Leaders of Higher Learning Institutions of Shanxi (TYAL) and a grant from National Foundation of China (No. 11001194).

References

  1. M. Kato, K.-S. Saito, and T. Tamura, β€œSharp triangle inequality and its reverse in Banach spaces,” Mathematical Inequalities & Applications, vol. 10, no. 2, pp. 451–460, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  2. M. Fujii, M. Kato, K.-S. Saito, and T. Tamura, β€œSharp mean triangle inequality,” Mathematical Inequalities & Applications, vol. 13, no. 4, pp. 743–752, 2010. View at Google Scholar
  3. K.-I. Mitani, K.-S. Saito, M. Kato, and T. Tamura, β€œOn sharp triangle inequalities in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1178–1186, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. K.-I. Mitani and K.-S. Saito, β€œOn sharp triangle inequalities in Banach spaces II,” Journal of Inequalities and Applications, vol. 2010, Article ID 323609, 17 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. S. Saitoh, β€œGeneralizations of the triangle inequality,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 3, article 62, pp. 1–5, 2003. View at Google Scholar Β· View at Zentralblatt MATH
  6. H. Belbachir, M. Mirzavaziri, and M. S. Moslehian, β€œq-norms are really norms,” The Australian Journal of Mathematical Analysis and Applications, vol. 3, no. 1, article 2, pp. 1–3, 2006. View at Google Scholar Β· View at Zentralblatt MATH
  7. K.-S. Saito, M. Kato, and Y. Takahashi, β€œVon Neumann-Jordan constant of absolute normalized norms on β„‚2,” Journal of Mathematical Analysis and Applications, vol. 244, no. 2, pp. 515–532, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. Y. Takahashi, M. Kato, and K.-S. Saito, β€œStrict convexity of absolute norms on β„‚2 and direct sums of Banach spaces,” Journal of Inequalities and Applications, vol. 7, no. 2, pp. 179–186, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH