Correspondence should be addressed to Wing-Sum Cheung,

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1 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2009, Article ID , 7 pages doi: /2009/ Research Article O Pečarić-Raić-Dragomir-Type Iequalities i Normed Liear Spaces Zhao Chagia, 1 Chur-Je Che, 2 ad Wig-Sum Cheug 3 1 Departmet of Iformatio ad Mathematics Scieces, College of Sciece, Chia Jiliag Uiversity, Hagzhou , Chia 2 Departmet of Mathematics, Tughai Uiversity, Taichug 40704, Taiwa 3 Departmet of Mathematics, The Uiversity of Hog Kog, Pokfulam Road, Hog Kog Correspodece should be addressed to Wig-Sum Cheug, wscheug@hku.hk Received 27 April 2009; Accepted 18 November 2009 Recommeded by Sever Silvestru Dragomir We establish some geeralizatios of the recet Pečarić-Raić-Dragomir-type iequalities by providig upper ad lower bouds for the orm of a liear combiatio of elemets i a ormed liear space. Our results provide ew estimates o iequalities of this type. Copyright q 2009 Zhao Chagia et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. 1. Itroductio I the recet paper 1, Pečarić ad Raić proved the followig iequality for ozero vectors k, k {1,...,} i the real or comple ormed liear space X, : 1 ma k k mi 1 k k 1.1 ad showed that this iequality implies the followig refiemet of the geeralised triagle

2 2 Joural of Iequalities ad Applicatios iequality obtaied by Kato et al. i 2: mi { k} ma { k}. 1.2 The iequality 1.2 ca also be obtaied as a particular case of Dragomir s result established i 3: ma 1 { } p1 p p p 1p { } mi p1 1 p, 1.3 where p 1ad 2. Notice that, i 3, a more geeral iequality for cove fuctios has bee obtaied as well. Recetly, the followig iequality which is more geeral tha 1.1 was give by Dragomir 4: ma α k α α k α mi α k α α k. 1.4 The mai aim of this paper is to establish further geeralizatios of these Pečarić-Raić- Dragomir-type iequalities 1.1, 1.2, 1.3,ad1.4 by providig upper ad lower bouds for the orm of a liear combiatio of elemets i the ormed liear space. Our results provide ew estimates o such type of iequalities.

3 Joural of Iequalities ad Applicatios 3 2. Mai Results Theorem 2.1. Let X, be a ormed liear space over the real or comple umber field K. If α K ad X for i 1,...,i {1,...,} with 2, the ma k {1,...,},..., { α k1,...,k } α α k1,...,k α 2.1 { mi α k1,...,k k {1,...,} i,..., 1 1 } α α k1,...,k. Proof. Observe that, for ay fied k {1,...,}, 1,...,, we have α α k1,...,k α α k1,...,k. 2.2 Takig the orm i 2.2 ad utilizig the triagle iequality, we have α α k 1,...,k α α k1,...,k α k1,...,k α α k1,...,k, 2.3 which, o takig the miimum over k iequality i 2.1. Net, by 2.2 we have obviously {1,...,}, 1,...,, produces the secod α α k1,...,k α k1,...,k α. 2.4

4 4 Joural of Iequalities ad Applicatios O utilizig the cotiuity property of the orm we also have α α k 1,...,k α α k1,...,k α k 1,...,k α α k1,...,k 2.5 α k1,...,k α α k1,...,k, which, o takig the maimum over k {1,...,}, 1,...,, produces the first part of 2.1 ad the theorem is completely proved. Remark 2.2. i I case the multi-idices i 1,...,i ad k 1,...,k reduce to sigle idices ad k, respectively, after suitable modificatios, 2.1 reduces to iequality 1.4 obtaied by Dragomir i 4. ii Furthermore, if X \{0} for {1,...,} ad α k 1/ k, k {1,...,} with 2, the iequality reduces further to iequality 1.1 obtaied by Pečarić ad Raić i1. iii Further to ii,if 2, writig 1 ad 2 y, we have y y mi {, } y y y y y ma {, y }, 2.6 which holds for ay ozero vectors, y X. The first iequality i 2.6 was obtaied by Mercer i 5. The secod iequality i 2.6 has bee obtaied by Maligrada i 6. It provides a refiemet of the Massera-Schäffer iequality 7: y y 2 y ma {, }, y 2.7 which, i tur, is a refiemet of the Dukl-Williams iequality 8: y y 4 y y. 2.8

5 Joural of Iequalities ad Applicatios 5 Theorem 2.3. Let X, be a ormed liear space over the real or comple umber field K. If α 1,..., K ad 1,..., X \{0} for 1,..., {1,...,} with 2, the 1 ma k1,...,k 1,..., k i {1,...,} i1,..., ,..., 1,..., i1,..., ,..., k1,...,k mi k i {1,...,} k1,...,k 1,..., 1 1 1,..., k1,...,k. This follows immediately from Theorem 2.1 by requirig 1,..., / 0for i 1,...,, ad lettig α k1,...,k 1/ k1 k for k i 1,...,; 2. A somewhat surprisig cosequece of Theorem 2.3 is the followig versio. Theorem 2.4. Let X, be a ormed liear space over the real or comple umber field K. If 1,..., X \{0} for 1,..., {1,...,} with 2, the 1,..., 1,..., mi 1 1,...,,..., ,..., i 1,..., i1,..., 1,..., 1,..., ma 1 1,...,,...,. 1 i 1,..., i1,..., Proof. Lettig ma i 1,...,, i1,..., 1,..., ad by usig the secod iequality i 2.9, we have 1,..., 1 1,..., 1,..., 1,..., ,..., Hece 1,..., 1,..., 1,..., ,..., ,...,

6 6 Joural of Iequalities ad Applicatios The it follows that 1,..., 1,..., 1,..., i1 1,...,,...,i ,..., ,..., ma ,...,,...,. i 1,..., i1,..., O the other had, lettig k1,...,k mi i 1,...,, i1,..., 1,..., ad by usig the first iequality i 2.9, we have 1,..., 1 1,..., k1,...,k 1,..., ,..., k1,...,k k1,...,k 1,..., k1,...,k 1 1,..., Hece k1,...,k 1,..., 1,..., 1,..., k1,...,k ,...,, from which we get 1,..., 1,..., 1,..., k1 1,...,,...,k ,..., ,..., mi ,..., 1,...,. i 1,..., i1,..., This completes the proof. Remark 2.5. I case the multi-idices 1,..., ad k 1,...,k reduce to sigle idices ad k, respectively, after suitable modificatios, 2.10 reduces to iequality 1.2 obtaied i 2 by Kato et al.

7 Joural of Iequalities ad Applicatios 7 Theorem 2.6. Let X, be a ormed liear space over the real or comple umber field K. If 1,..., X \{0} for 1,..., {1,...,} with 2 ad p 1, the mi 1 i i1,..., { } 1,..., p 1,..., p1 1,..., 1,..., p 1,..., p 1p 1,..., p { } ma 1,..., 1,..., p1 1,...,. 1 i i1,..., ,..., This follows much i the lie as the proofs of Theorem 2.1 ad Theorem 2.4, adsoit is omitted here. Remark 2.7. I case the multi-ide 1,..., reduces to a sigle ide, after suitable modificatios, 2.17 reduces to iequality 1.3 obtaied by Dragomir i 3. Ackowledgmets The first author s work is supported by the Natioal Natural Scieces Foudatio of Chia The third author s work is partially supported by the Research Grats Coucil of the Hog Kog SAR, Chia Proect o. HKU7016/07P. Refereces 1 J. Pečarić ad R. Raić, The Dukl-Williams iequality with elemets i ormed liear spaces, Mathematical Iequalities & Applicatios, vol. 10, o. 2, pp , M. Kato, K.-S. Saito, ad T. Tamura, Sharp triagle iequality ad its reverse i Baach spaces, Mathematical Iequalities & Applicatios, vol. 10, o. 2, pp , S. S. Dragomir, Bouds for the ormalised Jese fuctioal, Bulleti of the Australia Mathematical Society, vol. 74, o. 3, pp , S. S. Dragomir, Geeralizatio of the Pečarić-Raić iequality i ormed liear spaces, Mathematical Iequalities & Applicatios, vol. 12, o. 1, pp , P. R. Mercer, The Dukl-Williams iequality i a ier product space, Mathematical Iequalities & Applicatios, vol. 10, o. 2, pp , L. Maligrada, Simple orm iequalities, The America Mathematical Mothly, vol. 113, o. 3, pp , J. L. Massera ad J. J. Schäffer, Liear differetial equatios ad fuctioal aalysis. I, Aals of Mathematics, vol. 67, pp , C. F. Dukl ad K. S. Williams, A simple orm iequality, The America Mathematical Mothly, vol. 71, o. 1, pp , 1964.