on the triangle inequality in normed spaces

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1、OnthetriangleinequalityinnormedspacesKichi-SukeSaitoDepartmentofMathematicsFacultyofScienceNiigataUniversitysaito@math.sc.niigata-u.ac.jp1.IntroductionThetriangleinequalityisundoubtedlyoneofthemostfundamentalinequal-itiesinmathematics.LetXbeanormed(Banac

2、h)space.Foranyvectorsx;y2X,jjx+yjj·jjxjj+jjyjj(Triangleinequality):Severalauthorshavebeentreatingitsgeneralizationsandreverseinequali-ties(cf.Hudzik–Landes[7],S.Saitoh[16],Dragomir[2]andetc).Recently,Kato-Saito-Tamura[9]foundthesharptriangleinequalityand

3、itsreversein-equalitywithnelementsinanormedspacetostudythegeometricalstructureofBanachspaces.Afterthat,wehaveseveralpapersaboutthetrianglein-equalities(cf.J.Peˇcari´c–R.Raji´c[15],Dragomir[3,4]andHsu–Shaw–Wong[6]).Veryrecently,Mitani-Saito-Kato-Tamura[13

4、]provedtherefinementofsharptriangleinequalityandthereverseinequality.Ouraiminthistalkistopresenttherecentresultsofsharptrianglein-equalitiesin[8,13].2.SharptriangleinequalitiesandthereverseAtfirst,weconsidertwonon-zerovectorsx;yofanormedspaceX.Thenwehave1T

5、heorem1Fortwonon-zerovectorsx;y2Xsuchthatkxk¸kyk,µ°°¶°xy°kx+yk+2¡°°+°°kykkxkkyk(1)·kxk+kykµ°°¶°xy°(2)·kx+yk+2¡°°+°°kxk:kxkkykThefirstinequalitywithtwoelements(1)wasgivenearlierinHudzikandLandes[7];theinequalites(1)and(2)arealsofoundinarecentpaperofMaligra

6、nda[10].Wenextconsiderthreenon-zerovectorsx;y;zofanormedspaceX.ThenwehaveTheorem2.Forallnonzeroelementsx;y;zinaBanachspaceXwithkxk¸kyk¸kzk,µ°°¶°xyz°kx+y+zk+3¡°°++°°kzkkxkkykkzkµ°°¶°xy°+2¡°°+°°(kyk¡kzk)kxkkyk·kxk+kyk+kzkµ°°¶°xyz°·kx+y+zk+3¡°°++°°kxkkxkkyk

7、kzkµ°°¶°yz°¡2¡°°+°°(kxk¡kyk):kykkzkIngeneral,wehavethefollowingtriangleinequalitiesfornnonzerovectorsx1;¢¢¢;xn2X.Theorem3([13]).Letn¸3.Foranynon-zerovectorsx1;¢¢¢;xnofanormedspaceX,°°Xn°°Xnµ°°Xk¤°°¶°°°xj°¤¤°xj°+k¡°kx¤k°(kxkk¡kxk+1k)j=1k=2j=1j2Xn·kxjkj=1°

8、°Xn°°Xnµ°°Xn¤°°¶°°°xj°¤¤·°xj°¡k¡°kx¤k°(kxn¡kk¡kxn¡(k¡1)k);j=1k=2j=n¡(k¡1)jwherex¤;x¤;¢¢¢;x¤aretherearrangementofx;x;¢¢¢;xsatisfyingkx¤k¸12n12n1kx¤k¸¢¢¢¸kx¤k,andx¤=x¤=0.2n0n+1InTheorem3,wemayassumethatkx1k¸kx2k¸¢¢¢¸kxnk:tha