In this paper we obtain inequalities involving the norms of finite sequences of vectors in normed linear spaces, which are new even for complex numbers

Cho, Yeol Je

Dragomir, Sever S

Koliha, Jerry J

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Victoria University Institutional Repository (VUIR)

On Inequalities in Normed Linear Spaces and Applications

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Victoria University Eprints Repository

ON INEQUALITIES IN NORMEDLINEAR SPACES AND APPLICATIONSS. S. Dragomir, J. J. Koliha and Y. J. ChoAbstract. In this paper we obtain inequalities involving the norms of finite se-quences of vectors in normed linear spaces, which are new even for complex num-bers.I. IntroductionLet (X, ‖ · ‖) be a normed linear space. The following inequality is well knownin the literature as the polygonal inequality, which is a generalization of thetriangle inequality:(P)∥∥∥∑i∈Ixi∥∥∥ ≤∑i∈I‖xi‖;I is a finite set of indices and xi (i ∈ I) are vectors in X.In papers [1–3], authors explored relations between the following refinementsand generalizations of (P):Theorem A. Let (X, ‖ · ‖) be a normed linear space and I be a finite set ofindices, xi ∈ X, i ∈ I and p ∈ R, p ≥ 1. Then we have the following inequalities:(1)∥∥∥∑i∈Ixi∥∥∥p ≤ [card(I)]p−k−1 ∑i1,...,ik+1∈I∥∥∥∥xi1 + · · ·+ xik+1k + 1 ∥∥∥∥p≤ [card(I)]p−k ∑i1,...,ik∈I∥∥∥∥xi1 + · · ·+ xikk ∥∥∥∥p≤ · · ·≤ [card(I)]p−2 ∑i1,i2∈I∥∥∥xi1 + xi22∥∥∥p≤ [card(I)]p−1∑i∈I‖xi‖p,Typeset by AMS-TEX12 S. S. DRAGOMIR, J. J. KOLIHA AND Y. J. CHOwhere k ∈ N , k ≥ 1.The following result is a refinement of (P) for weighted mean:Theorem B. With the above assumptions, if gj (j = 1, 2, 3, . . . , k) are nonneg-ative weights such that∑kj=1 gj > 0, then we have the following inequality:(2)∥∥∥∑i∈Ixi∥∥∥p ≤ [card(I)]p−k ∑i1,...,ik∈I∥∥∥∥xi1 + · · ·+ xikk ∥∥∥∥p≤ [card(I)]p−k ∑i1,...,ik∈I∥∥∥∥g1xi1 + · · ·+ gkxikg1 + · · ·+ gk∥∥∥∥p≤ [card(I)]p−1∑i∈I‖xi‖p.Note that both the above inequalities were obtained from more general resultsfor convex mappings defined on convex subsets in normed linear spaces.In this paper, using only techniques from normed linear spaces, we obtain newinequalities for the norms of finite sequences.II. The ResultsWe start with the following theorem:Theorem 2.1. Let (X, ‖ · ‖) be a normed linear space, xi ∈ X with i ∈ I (I isfinite) and x =∑i∈I xi. Then we have the following inequalities:(3)∥∥∥‖x‖x−∑i∈I‖x± xi‖xi∥∥∥ ≤∑i∈I‖xi‖2 ≤ 12(‖x‖+∑i∈I‖xi‖) ∑i∈I‖xi‖.Proof. For a fixed i ∈ I, we have‖xi‖ =∥∥∥x− ∑j∈I,j 6=ixj∥∥∥ ≤ ‖x‖+ ∥∥∥ ∑j∈I,j 6=ixj∥∥∥≤ ‖x‖+ ∑j∈I,j 6=i‖xj‖ = ‖x‖+∑j∈I‖xj‖ − ‖xi‖,which implies that, for all i ∈ I,‖xi‖ ≤ 12(‖x‖+∑j∈I‖xj‖).ON INEQUALITIES IN NORMED LINEAR SPACES 3Multiplying by ‖xi‖ ≥ 0 and summing over i ∈ I, we deduce∑i∈I‖xi‖2 ≤ 12(‖x‖+∑j∈I‖xj‖) ∑i∈I‖xi‖,and so the second inequality in (1) is proved. Also, for a fixed i ∈ I, we have‖xi‖ = ‖ − xi‖ = ‖x± xi − x‖ ≥∣∣∣‖x± xi‖ − ‖x‖∣∣∣.Multiplying by ‖xi‖ ≥ 0, we have‖xi‖2 ≥∣∣∣‖x± xi‖ − ‖x‖∣∣∣ ‖xi‖ = ∥∥∥(‖x± xi‖ − ‖x‖)xi∥∥∥for all i ∈ I. Summing over i ∈ I, we have∑i∈I‖xi‖2 ≥∑i∈I∥∥∥‖x± xi‖xi − ‖x‖xi∥∥∥ ≥ ∥∥∥∑i∈I‖x± xi‖xi − ‖x‖xi∥∥∥,and so the theorem is proved.Corollary 2.2. With the above assumptions, if∑i∈I xi = 0, then(4)∑i∈I‖xi‖2 ≤ 12(∑i∈I‖xi‖)2.The following result generalizes Theorem 2.1. In its proof we need the formula(5)∑i1,...,ik∈I(xi1 + · · ·+ xik) = k[card(I)]k−1∑i∈Ixi,which can be proved by induction on k.Theorem 2.3. With the above assumptions, we have the following inequalities:(6)k∥∥∥ ∑i1,...,ik∈I‖x± (xi1 + · · ·+ xik)‖xi1 − nk−1‖x‖x∥∥∥≤ ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2≤ 12( ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2 + k∑i1,...,ik∈I‖xi1‖ ‖xi1 + · · ·+ xik‖)≤ 12(‖x‖+∑i∈I‖xi‖) ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖,4 S. S. DRAGOMIR, J. J. KOLIHA AND Y. J. CHOwhere k ≥ 1 and card(I) = n.Proof. For fixed i1, . . . , ik ∈ I, we have‖xi1 + · · ·+ xik‖ =∥∥∥x− ∑j /∈{i1,...,ik}xj∥∥∥≤ ‖x‖+∥∥∥ ∑j /∈{i1,...,ik}xj∥∥∥≤ ‖x‖+ ∑j /∈{i1,...,ik}‖xj‖= ‖x‖+∑i∈I‖xi‖ − ‖xi1‖ − · · · − ‖xik‖,from which it follows‖xi1 + · · ·+ xik‖+ ‖xi1‖+ · · ·+ ‖xik‖ ≤ ‖x‖+∑i∈I‖xi‖.Multiplying by ‖xi1 + · · ·+ xik‖ ≥ 0 and summing over i1, . . . , ik in I, we have∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2 + k∑i1,...,ik∈I‖xi1‖ ‖xi1 + · · ·+ xik‖≤(‖x‖+∑i∈I‖xi‖) ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖,and so the third inequality in (6) follows. By the triangle inequality‖xi1 + · · ·+ xik‖ ≤ 12 (‖xi1 + · · ·+ xik‖+ ‖xi1‖+ · · ·+ ‖xik‖).Multiplying by ‖xi1 + · · ·+ xik‖ and summing over i1, . . . , ik in the index set I,we derive∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2≤ 12( ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2 + k∑i1,...,ik∈I‖xi1‖‖xi1 + · · ·+ xik‖)and so the second inequality in (6) is also proved.ON INEQUALITIES IN NORMED LINEAR SPACES 5Finally, we prove the first inequality in (6). Since‖xi1 + · · ·+ xik‖ = ‖ − (xi1 + · · ·+ xik)‖= ‖x± (xi1 + · · ·+ xik)− x‖≥∣∣∣‖x± (xi1 + · · ·+ xik)‖ − ‖x‖∣∣∣,then‖xi1 + · · ·+ xik‖2 ≥∥∥∥(‖x± (xi1 + · · ·+ xik)‖ − ‖x‖)(xi1 + · · ·+ xik)∥∥∥=∥∥∥‖x± (xi1 + · · ·+ xik)‖(xi1 + · · ·+ xik)− ‖x‖(xi1 + · · ·+ xik)∥∥∥.Summing over i1, . . . , ik in the index set I, we use (5) to obtain∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2≥ ∑i1,...,ik∈I∥∥∥‖x± (xi1 + · · ·+ xik)‖(xi1 + · · ·+ xik)− ‖x‖(xi1 + · · ·+ xik)∥∥∥≥∥∥∥k ∑i1,...,ik∈I‖x± (xi1 + · · ·+ xik)‖xi1 − knk−1‖x‖∑i∈Ixi∥∥∥= k∥∥∥ ∑i1,...,ik∈I‖x± (xi1 + · · ·+ xik)‖xi1 − nk−1)‖x‖x∥∥∥and thus the theorem is proved.Corollary 2.4. With the above assumptions, if∑i∈I xi = 0, then(7)k∥∥∥ ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖xi1∥∥∥≤ ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2≤ 12( ∑i1,...,ik∈I‖xi1 + · · ·+ xik‖2 + k∑i1,...,ik∈I‖xi1‖ ‖xi1 + · · ·+ xik‖)≤ 12∑i∈I‖xi1‖∑i1,...,ik∈I‖xi1 + · · ·+ xik‖.III. Applications to Complex Numbers6 S. S. DRAGOMIR, J. J. KOLIHA AND Y. J. CHOThe inequalities obtained in Section II can be restated to obtain new inequal-ities for complex numbers:1. Let zi ∈ C, i ∈ I (I is finite) and let z =∑i∈I zi. Then∣∣∣|z|z −∑i∈I|z ± zi|zi∣∣∣ ≤∑i∈I|zi|2 ≤ 12(|z|+∑i∈I|zi|) ∑i∈I|zi|.In particular, if z = 0, then∑i∈I|zi|2 ≤ 12(∑i∈I|zi|)2.2. With the above assumptions, we havek∣∣∣ ∑i1,...,ik∈I|z ± (zi1 + · · ·+ zik)|zi1 − nk−1|z|z∣∣∣≤ ∑i1,...,ik∈I(zi1 + · · ·+ zik)2≤ 12( ∑i1,...,ik∈I|zi1 + · · ·+ xik |2 + k∑i1,...,ik∈I|zi1 | |zi1 + · · ·+ zik |)≤ 12(|z|+∑i∈I|zi|) ∑i1,...,ik∈I|zi1 + · · ·+ zik |,where k ∈ N , k ≥ 1 and card(I) = n.In particular, if z = 0, we havek∣∣∣ ∑i1,...,ik∈I|zi1 + · · ·+ zik |zi1∣∣∣ ≤ ∑i1,...,ik∈I|zi1 + · · ·+ zik |2≤ 12( ∑i1,...,ik∈I|zi1 + · · ·+ xik |2 + k∑i1,...,ik∈I|zi1(zi1 + · · ·+ zik)|)≤ 12∑i∈I|zi|∑i1,...,ik∈I|zi1 + · · ·+ zik |.References1. S. S. Dragomir, Some refinements of Ky Fan’s inequality, J. Math. Anal. Appl. 163 (1992),317–321.2. S. S. Dragomir, On some refinements of Jensen’s inequality and applications, Utilitas Math.43 (1993), 235–243.3. J. E. Pecaric and S. S. Dragomir, A refinement of Jensen’s inequality and applications,Studia Univ. Babes-Bolyai Math. 34 (1989), 15–19.ON INEQUALITIES IN NORMED LINEAR SPACES 7School of Communications and Informatics, Victoria University of Technol-ogy, Melbourne, VIC 8001, AustraliaDepartment of Mathematics and Statistics, University of Melbourne, VIC 3010AustraliaDepartment of Mathematics, Gyeongsang National University, Chinju 660-701,Korea

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On Inequalities in Normed Linear Spaces and Applications

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