GEOMETRY OF JENSEN S INEQUALITY AND QUASI-ARITHMETIC MEANS

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1 IJRRS ugust 3 wwwararesscom/volumes/volissue/ijrrs 4d GEOMETRY OF JENSEN S INEQULITY ND QUSI-RITHMETIC MENS Zlato avć Mechacal Egeerg Facult Slavos Brod Uverst o Osje Trg Ivae Brlć Mažurać 35 Slavos Brod Croata E-mal: Zlatoavc@ssbhr BSTRCT The aer observes the basc roertes o cove ucto the dscrete ad tegral case The coecto o covet wth uatt ceters s deted the dscrete case The relato betwee barceters ad tegral arthmetc meas s studed the tegral case The eualtes or uas-arthmetc meas are observed both cases Ke words ad hrases: Cove ucto rd ceter o the cove combato barcetre o the set uas-arthmetc mea Mathematcs Subject Classcato: 5 E 8 5 INTRODUCTION Through ths aer I wll be a terval wth the o-emt teror wll be deoted b I I characterstc ucto o a set X X ad cove hull o X wll be deoted b co X I are ots ad are coecets such that the the sum c belogs to I ad t s called the cove combato o I The umber c tsel s called the ceter o the cove combato For a cotuous ucto I Let : a b be a cove ucto ad : I the assocated cove combato a b belogs to be the rd le jog the ots a a B b b o the grah o Ever a b ca be reseted as the cove combato b a a b b a b a ad the cove ucto satses the eualt ad b a a b a b b a b a o the grah o so that The olgoal rd le jog the ots < < wll be used as the ucto I lg : > { } the ever cove ucto veres the double eualt lg DISCRETE CSE The ma result ths secto s Theorem 5 whch reresets covet b usg the commo ceter o deret cove combatos Let be a lae For a ot the radus-vector O cosderg some ed ot O wll be deoted b r ad smlarl or a ot the radus-vector wll be deoted b r 3

2 IJRRS ugust 3 avć Geometr o Jese s Ieualt Theorem altc resetato o Cove olgo Let vertces radus-vector r o a ot ad ol Coseuetl veres the cove combato eualt r r be a cove olgo wth r r 3 Remar Cove combatos are uue ol or a le segmet have a uadragle B C D B D ad tragle 3 the we ca ose the ot that belogs to the tragles B C I we ad ad does ot belog to the edge B Usg the radus-vectors o the vertces o these tragles we have two deret our-membered cove combatos so that r r BrB CrC rd r BrB rc DrD > ad > The above cove combatos are deret because Theorem B Cove olgo resetato o Cove Fucto real valued ucto wth real varable s strctl cove ad ol ever -tule o the ots o the grah o deleates the cove olgo wth vertces Corollar Chord resetato o Cove Fucto ucto satses the double eualt or ever cove combato roo Suose s cove ad tae a cove combato ots C D : I s cove ad ol t lg o I wth < < rom I wth < < co bouded wth the rds lg r o the orm r r 4 deleate the cove olgo b Theorem B The ed-ot o a radus-vector Theorem So we have the cluso whch reresets the double eualt 4 Suose the eualt 4 s vald ad tae a bomal cove combato < < Usg the let had sde o the eualt 4 we get whch roves the covet o the ucto lg The double eualt 4 ca be eteded to the seres o eualtes The the ad belogs to C b rom I such that 4

3 IJRRS ugust 3 avć Geometr o Jese s Ieualt lg lg The secod eualt 5 s the well-ow Jese s eualt whch characterzes cove uctos that s a ucto s cove ad ol t satses Jese s eualt 5 Fgure : Grahcal resetato o the eualt 4 We reset brel the hscal meag o cove combatos Cosder a set o artcles ots the lae The value o a certa hscal uatt mass dest otetal s measured at each artcle o the observed set We wat to sec the ceter o the uatt Let the artcles be located at the ots o-egatve uatt values relatve uatt values ad ostve uatt total value tot / tot or uatt ceter s the cove combato o the gve osto vectors ccordgl we have wth So we ca tae the It s reasoable to assume that the radus-vector r o the r wth the coecets r r r tot Relg o Theorem t ca be cocluded the ceter s located the cove hull o the ots { } Ver ractcal descrtos o covet was reseted Let co I be ots ad < < or some s vald the t ollows that s be o-egatve umbers such that I the cove combato eualt 5

4 IJRRS ugust 3 avć Geometr o Jese s Ieualt The et lemma gves the coecto betwee uatt ceters ad Jese s te eualtes Lemma 3 Let I be ots such that } { } { co Let be o-egatve umbers such that < < I oe o the cove combato eualtes 7 s vald the the double cove combato eualt 8 holds or ever ucto R I : whch satses the double eualt 4 roo Let stads or all cove combatos 7 Wthout loss o geeralt suose that all are arwse deret ad < < I we al the rght had sde o the eualt 4 o the ots we get lso suose that } < < } { { The ot s act lg lg the let had sde o the eualt 4 o the ots we get lg Coectg the above eualtes we get The the bomal cove combato reresets the double eualt 8 Lemma 4 I a ucto I : satses the eualt 8 uder the codto 7 the t satses the Jese eualt roo Let Wthout loss o geeralt suose that all are arwse deret ad all > I or all the we have } { } { co We ca al the outer sde o the eualt 8 o the sets } { ad } { to get the Jese eualt cosderg

5 IJRRS ugust 3 avć Geometr o Jese s Ieualt I or some deret rom : Sce 9 the the ot ca be eressed aga as the cove combato o the ots or all we ca al the revous case ad obta the eualt rom whch ollows rragg the above eualt cosderg the assumto we get aga the Jese eualt 9 Theorem 5 Quatt Ceter resetato o Cove Fucto ucto t satses the eualt 8 uder the codto 7 : I s cove ad ol Eamle Lemma 3 does ot geerall hold or cove uctos o several varables as show the ollowg eamle the coordate lae Tae ots ad ucto 5 ad Let ad strct double eualt Let C co 3 4 r be the radus-vectors o the ots The suare C does ot cota the ots The we have the double eualt all members euals 4 r r > > ad because 4 r 5 5 I alcatos o covet we ote use strctl mootoe cotuous uctos cove wth resect to s -cove that s rom 5 Deto 9 smlar otato s used or cocavt Let s cove o I : I such that s ths termolog s tae be a cove combato o I The dscrete -uas-arthmetc mea o the ots artcles wth the coecets weghts s the ot whch belogs to I because the cove combato belogs to I 7

6 IJRRS ugust 3 avć Geometr o Jese s Ieualt Comlete the secto wth the alcato o Lemma 3 o uas-arthmetc meas Theorem 7 Let I be ots such that co{ } { } < be o-egatve umbers such that Let < Let : I be strctl mootoe cotuous uctos I s ether -cove ad creasg or -cocave ad decreasg ad oe o the eualtes s vald the the double eualt holds I s ether -cove ad decreasg or -cocave ad creasg the the reverse double eualt s vald roo Brel J I we al Lemma 3 wth the ots J ad cove or cocave ucto : J More geerall about the deret orms o uas-arthmetc meas ca be oud the artcle 3 INTEGRL CSE The ma result ths secto s Theorem 35 whch reresets covet b usg the commo barceter o deret tervals The tegral varats o Jese s te eualtes are also obtaed b usg the barceters We wll use a te measure o I assumg that all subtervals o I are -measurable Itegral aalog o the cocet o cove combato s the cocet o barceter Let wth > sets I be a -measurable Gve a ostve teger let be a artto o arwse dsjot -measurable be ots The we have the cove combato ad whose ceter b c belogs to co I the seuece c c coverges the the -barceter o ca be deed lm d 3 : I s -tegrable o the Itegral arthmetc mea o a ucto s smlarl deed I a ucto the -arthmetc mea o o s deed b lm d 4 Note that where deotes the dett ucto o I s the terval the ts -barceter belogs to ad s cotuous o the ts -arthmetc mea o belogs to 8

7 IJRRS ugust 3 avć Geometr o Jese s Ieualt The basc rule or summato o barceters ad tegral arthmetc meas sas: roosto Let be a te measure o I ad sets I wth > I the -barceter I a ucto ests the : I s -tegrable o the be a uo o arwse dsjot -measurable 5 Itegral aalog o Corollar s oe varat o the Hermte-Hadamard eualt whch shows the et corollar Corollar 3 Let be a te measure o I ad I s cove ad cotuous the t satses the double eualt : I be a ucto a b d a b a b a b a b wth a b > ad ts -barceter / a b d a b or ever bouded closed terval I a b roo Gve a ostve teger let ever cotracts to the ot as goes to t For ever / a b < < a b be a artto o arwse dsjot tervals 7 where we tae oe ot also tae Suose that lg the eualt 4 or ths case we have The ollowg lmts lg ad 8 lm lm lm lm lg d a b a b a b a b d a b a b a b a b a b hold so the eualt 8 "coverges" to the eualt 7 Basc coectos betwee the covet ad the Hermte-Hadamard eualt ca be oud 3 ages 5-53 Now wee wll use the te measure o I whch s ostve o the tervals that s whch satses > or ever o-degeerate terval I 9

8 IJRRS ugust 3 avć Geometr o Jese s Ieualt Usg the tegral method wth cove combatos as the revous corollar we get the tegral varat o Lemma 3: Lemma 3 Let be a te measure o I that s ostve o tervals Let B ad < B I oe o the barceter eualtes B I be tervals so that B 9 s vald the the double tegral arthmetc mea eualt B holds or ever -tegrable cove ucto roo Gve a ostve teger let dsjot tervals where ever : I B be a artto wth the subartto o arwse cotracts to the ot or vashes t as goes to t ut Sce we suose the barceter eualt 9 s vald we ca ose the ots that sats the ceter eualt B B \ For eamle we ca ut / d So we ca al the eualt 8 ad get B Lettg to t we obta the eualt B \ The varat o Lemma 3 or the bouded closed tervals ad B was roved 4 roosto b usg the rd le a b Lemma 3 s ot geerall true or cove uctos o several a b the case varables as ca be see 4 Eamles ad Remar 33 Lemma 3 s also vald the case whe the set B s a uo o tervals measure o I s sad to be cotuous {t} or ever ot t I accordg to the deto the boo age 49 Lemma 34 Let be a cotuous te measure o I that s ostve o the tervals I a I s a ot the the decreasg seres roo Tae a ot I the rst ste we ose the ots terval : I a a o bouded tervals I ests so that a ad a I ad show the rst two stes I the we tae { a} such that < a < a t a > ad determe the -barceter o the td I a g a td t a the we observe the ucto g : deed b

9 IJRRS ugust 3 avć Geometr o Jese s Ieualt The ucto g s strctl creasg cotuous wth g < ad g a > Thereore a < a < a the ucto g has the uue zero-ot > act > I a < a we use the ucto g a I the et ste we tae the ots ad reeat the revous rocedure to determe a a ad I ths case we tae Theorem 35 Barceter resetato o Cove Fucto Let be a cotuous te measure o I that s ostve o the tervals -tegrable cotuous ucto : I s cove ad ol t satses the eualt uder the codto 9 roo Necesst s roved Lemma 3 Now let us rove the sucec Tae a cove combato al Lemma 34 to determe the decreasg seres ever barceter a a rom I wth < ad < > The a I so we ca whch cotracts to {a} ad o bouded tervals I a smlar wa or ever t s ossble to determe the uo so that a ad that ad as B ad The the eualt resectvel B \ holds or ever Sce ad lm Lettg td t td t B \ a lm ad lm lg the outer sde o the eualt o the ar o the sets t d t B \ B \ ad usg the cotut o we acheve a whch roves the covet o because td t lm Let the t d t t d t a ad B we get We eed the ollowg geeralzato o Lemma 3 or alcatos o uas-arthmetc meas t d t Corollar 3 Let be a te measure o I that s ostve o the tervals Let B I that B ad < B Let g : I be a -tegrable cotuous ucto ad g I I oe o the tegral arthmetc mea eualtes be tervals so J

10 IJRRS ugust 3 avć Geometr o Jese s Ieualt g g B g s vald the the double tegral arthmetc mea eualt g g B g holds or ever cove ucto : J rovded that g s -tegrable Let be a te measure o I I be -measurable set wth > ad : I mootoe cotuous ucto that s -tegrable o The tegral -uas-arthmetc mea o the set wth resect to the measure s the umber I s the terval the s ot coected the d be a strctl 3 belogs to because / d belogs to ma be outsde o I I I deotes the dett ucto o I the I -uas-arthmetc mea o s just the -barceter o that s / d I Theorem 37 Let be a te measure o I that s ostve o the tervals Let B I B ad < B Let : I be -tegrable strctl mootoe cotuous uctos I s ether -cove ad creasg or -cocave ad decreasg ad oe o the eualtes s vald the the double eualt be tervals so that B 4 B 5 holds I s ether -cove ad decreasg or -cocave ad creasg the the reverse double eualt s vald 5 roo Let us rove the case whe s -cove ad creasg Frst we al the ucto o the eualtes 4 ad get ut I d B d B d J Now we ca al Corollar 3 wth cove ucto : J R we have d d d B B B \ Fall we al the creasg ucto ad sce o the above double eualt ad get the double eualt 5 Reereces J M c c Z av c ad J e c ar c "The eualtes or uasarthmetc meas" bstract ad led alss vol rtcle ID ages T Needham " vsual elaato o Jese s eualt" merca Mathematcal Mothl vol C Nculescu ad L E ersso Cove Fuctos ad Ther lcatos Caada Mathematcal Socet Srger New Yor US 4 Z av c J e c ar c ad I er c "Itegral dscrete ad uctoal varats o Jese s eualt" Joural o Mathematcal Ieualtes vol 5 o J E e c ar c F roscha ad Y L Tog Cove Fuctos artal Ordergs ad Statstcal lcatos cademc ress New Yor US 99 W Rud Real ad Comle alss McGraw-Hll New Yor US 987