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MORE ON REVERSE TRIANGLE INEQUALITY ININNER PRODUCT SPACESA. H. ANSARI AND M. S. MOSLEHIANReceived 8 February 2005 and in revised form 17 May 2005Refining some results of Dragomir, several new reverses of the generalized triangle in-equality in inner product spaces are given. Among several results, we establish some re-verses for the Schwarz inequality. In particular, it is proved that ifais a unit vector ina real or complex inner product space (H;
·,·),r,s>0,p∈(0,s],D={x∈H,rx−sa≤p},x1,x2∈D−{0},andαr,s=min{(r2xk2−p2+s2)/2rsxk:1≤k≤2},then(x1x2−Re
x1,x2)/(x1+x2)2≤αr,s.1. IntroductionIt is interesting to know under which conditions the triangle inequality went the otherway in a normed spaceX; in other words, we would like to know if there is a positiveconstantcwith the property thatc
nk=1xk≤
nk=1xkfor any finite setx1,...,xn∈X.Nakai and Tada [7] proved that the normed spaces with this property are precisely thoseof finite dimensional.The first authors investigating reverse of the triangle inequality in inner product spaceswere Diaz and Metcalf [2] by establishing the following result as an extension of an in-equality given by Petrovich [8] for complex numbers.Theorem1.1 (Diaz-Metcalf theorem).Letabe a unit vector in an inner product space(H;
·,·).Supposethevectorsxk∈H,k∈{1,...,n}satisfy0≤r≤Rexk,axk,k∈{1,...,n}.(1.1)Thenrnk=1xk≤nk=1xk,(1.2)Copyright©2005 Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical Sciences 2005:18 (2005) 2883–2893DOI:10.1155/IJMMS.2005.2883
2884 Reverse triangle inequalitywhere equality holds if and only ifnk=1xk=rnk=1xka.(1.3)Inequalities related to the triangle inequality are of special interest (cf. [6,ChapterXVII]). They may be applied to get interesting inequalities in complex numbers or tostudy vector-valued integral inequalities [4,5].Using several ideas and following the terminology of [4,5], we modify or refine someresults of Dragomir and ours [1] and get some new reverses of triangle inequality. Amongseveral results, we show that ifais a unit vector in a real or complex inner productspace (H;
·,·),xk∈H−{0},1≤k≤n,α=min{xk:1≤k≤n},p∈(0,√α2+1),max{xk−a:1≤k≤n}≤p,andβ=min{(xk2−p2+1)/2xk:1≤k≤n},thennk=1xk−nk=1xk≤1−ββRenk=1xk,a.(1.4)We also examine some reverses for the celebrated Schwarz inequality. In particular, it isproved that ifais a unit vector in a real or complex inner product space (H;
·,·),r,s>0,p∈(0,s],D={x∈H,rx−sa≤p},x1,x2∈D−{0},andαr,s=min{(r2xk2−p2+s2)/2rsxk:1≤k≤2},thenx1x2−Rex1,x2x1+x22≤αr,s.(1.5)Throughout the paper, (H;
·,·) denotes a real or complex inner product space. Weuse repeatedly the Cauchy-Schwarz inequality without mentioning it. The reader is re-ferred to [3,9] for the terminology on inner product spaces.2. Reverse of triangle inequalityWe start this section by pointing out the following theorem of [1] which is a modificationof [5,Theorem3].Theorem2.1.Leta1,...,ambe orthonormal vectors in the complex inner product space(H;
·,·).Supposethatfor1≤t≤m,rt,ρt∈R, and that the vectorsxk∈H,k∈{1,...,n}satisfy0≤r2txk≤Rexk,rtat,0≤ρ2txk≤Imxk,ρtat,1≤t≤m.(2.1)Thenmt=1r2t+ρ2t1/2nk=1xk≤nk=1xk(2.2)