ON JENSEN S AND HERMITE-HADAMARD S INEQUALITY

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1 IJRRAS 7 3 Deemer 203 wwwrressom/volumes/vol7issue3/ijrras_7_3_02 ON JENSEN S AND HERMITE-HADAMARD S INEQUALITY Zlto Pvć & Ver Novosel 2 Mehl Egeerg Fulty Slvos Bro Uversty o Osje Trg Ive Brlć Mžurć Slvos Bro Crot Eml: ZltoPv@sshr 2 Mehl Egeerg Fulty Slvos Bro Uversty o Osje Trg Ive Brlć Mžurć Slvos Bro Crot Eml: VerNovosel@sshr ABSTRACT The rtle els wth the geerlztos o Jese s equlty the srete tegrl orm Oe geerlzto o Hermte-Hmr s equlty s lso resete The trsto rom srete to tegrl mes s relze usg the tegrl metho wth ove omtos Keywors: 26A5 26D5 28A25 52A40 ove omto e omto ove uto Jese s equlty Hermte-Hmr s equlty INTRODUCTION The tegrl equltes hve sel le the rh o mthemtl equltes Wth etesve mesure theory these equltes e wely use mthemts hyss Beses usg srete mesures tegrl equltes e esly resete the srete orm The reverse roeure tht strts rom the srete se roues the tegrl lso e mlemete y lyg the tegrl metho wth ove omtos The ltter s more ult Rell the two well-ow wely use tegrl equltes or rel vlue ove uto ee o oue lose tervl o rel umers Every ove uto :[ ] veres the equlty ow s Hermte-Hmr s equlty the equlty ow s the tegrl orm o Jese s equlty 2 2 Hermte-Hmr s equlty reers to the estmto o the tegrl rthmet me o ove uto I 883 Hermte hs sovere the oule equlty whh ws ulshe [3] Te yers lter Hmr hs resovere the let-h se o ths equlty whh ws resete [2] More etls o ths rmt story see [7] or [8 ges 62-63] I the srete se Jese s equlty s le to ove omtos o vetors rom ove set I the tegrl se Jese s equlty s wely le to ryeters tegrl mes I 906 Jese hs rove the verso o the equlty 2 usg Leesgue mesure whh ws ulshe [4] 2 INEQUALITIES WITH AFFINE COMBINATIONS Ths seto s rere org to Theorem A whh s oe o the m results o the sumtte er [0] Cove sets re geerlly oserve rel vetor se Alto to some vetor set s lytlly eresse y omtos o vetors ots The sum slrs oeets 2 263

2 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty elogs to the vetor suse l{ } the smllest vetor se tht ots ll t s lle the ler omto I tht ots ll the sum 3 elogs to the e hull { } the smllest trslte vetor se t s lle the e omto I ll [0] o{ the sum 3 elogs to the ove hull } the smllest ove vetor set tht ots ll t s lle the ove omto I wht ollows we use rel tervl [ ] ssumg < Every e uquely resete s the e omto 4 where Let : e ove uto B o the grh o I [ ] equlty I [ ] ho e the hor le ssg through the ots 5 { } A the the ove omto s ove we hve the hor ho { 6 the the reverse equlty s vl 6 ho Smlty o the hor le { } s the e uto h l wll e very useul to our wor For ths reso here s the ollowg lemm: Lemm 2 Let h :R R veres the equlty } e ots Let e oeets o the sum h h We wll eme the ehvor o ove utos o some sel tyes o e omtos Lemm 22 Let [ ] The every ove uto The every e uto e ot Let [0] [] e oeets o the sum :[ ] veres the equlty Proo Frst o ll let us show tht the e omto to e or oeets 8 elogs to [ ] Se [ ] te rom the ormuls 5 The we hve [ ] [ ] t hs The oeets squre rets re o-egtve wth the sum equls so the oserve eresso elogs to [ ] I 0 the equlty 8 s the Jese equlty or the three-memere ove omto

3 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty I 0 we use the equlty 6 the ty o the hor le ho { } ths wy: ho { } ho ho ho { } { } { } ho ho resetg tht [ ] { } ho { } Suet otos o the oeets Lemm 22 re: [0] otos ollows [] From these Fgure : Geometr terretto o the equlty 8 The lyt equlty wrtte the ormul 8 e esre y geometr gure Gve [ ] te the grh ots A B C eterme the osto o the ots P 0 or [0] We hve the rus-vetors equlty rp ra rb rc ra rb ra rb rc I 0 the let-h se o the equlty reresets the ove omtos o the vetors r A r B r C Iterrete geometrlly the ots P elog to the trgle o{ A B C} I 0 the oeets [0] wth the sum equl to Te the ot D The rght-h se o the equlty reresets the ove r A r B r D Ths mes tht the geometr loto o the ots P s just the o{ A B D So gve uto ot [ ] oeets o the sum the ots omtos o the vetors trgle } o the ormul 0 elog to the ove qurgle llustrte Fgure oa C D B 265

4 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty Theorem A Let [ ] e ots Let [0] [] The every ove uto :[ ] e oeets o the sums veres the equlty 2 For 0 the equlty 2 s reue to the Jese-Merer equlty see [5] Corollry 23 Let [ ] the equlty 2 s reue to the Jese equlty For e ots Let [0] 0 The every ove uto :[ ] R Proo Put suose < e oeets o the sum veres the equlty 3 The the e omto oes wth tht o Theorem A euse / 0 The equlty 3 ow ollows ter lyg Theorem A Remr 24 Let otg tht Suose tht e e omto wth the smllest ot the lrgest ot where [0] I 2 0] 4 2 [ wth > 0 2 < 0 the the oserve e omto elogs to the tervl { } It s true euse the ove omtos elog to { } Puttg we hve o / 2 / / 2 / o / 0 / 0 tht s the e omto wth the oeets The omto o{ } y Lemm 22 the sme s true or the omto

5 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty 4 3 TRANSITION FROM CONVEX COMBINATIONS TO INTEGRALS The turl wy o trsto rom srete to tegrl mes s oe tht volves ove omtos the tegrl metho Itegrl logy o the oet o ove omtos s the oet o ryeter Let us show how the ove omtos e mlemete to the tegrl metho Let e mesure ostve mesure org to [] o set S wth S > 0 S e rtto o rwse sjot -mesurle sets S s o the ove omto eters overges we get the set ryeter s the lmt I S s S lm B lm s Gve ostve teger let S e ots I the sequee So the -ryeter o S e ee wth : s ether o-egtve or o-ostve uto wth 0 the -ryeter o o S e ee wth B lm lm S rove tht the lmt ests Note tht the ove tegrl sum s the -memere ove omto o the ots wth o-egtve oeets I S s ove our se the tervl the B B elog to I ths wy t s ossle to me the trsto rom srete orm o Jese s equlty to the tegrl orm S S 7 S S I g : s -tegrle uto the -rthmet me o g o S s ee wth g M g lm g 8 rove tht the lmt ests I g s the otuous uto o the tervl ts -rthmet me elogs to

6 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty g 4 APPLICATIONS The m result ths seto s Theorem 45 whh etes the Hermte-Hmr equlty to e omtos rom the tervl The srete equlty 2 e le to ow tegrl equltes metoe Seto to tegrl qus-rthmet mes 4 Alto to Jese s Iequlty Alyg the equlty 2 wth the ove omtos o the eto 6 we hve the ollowg result: Corollry 4 Let e mesure o the tervl [ ] Let :[ ] e ether o-egtve or oostve -tegrle uto wth 0 Let [0] [] sum The the equlty e oeets o the 9 hols or every ove uto :[ ] rove tht the utos re - tegrle The equlty 7 s stll vl we susttute the etty uto wth uto equlty 9 e etee usg the uto g tht hs etreme vlues: g :[ ] The the Corollry 42 Let e mesure o the tervl [ ] Let :[ ] e ether o-egtve or oostve -tegrle uto wth 0 Let g :[ ] e -mesurle uto wth etreme vlues g g so g o{ g g } I or every [ ] hols or every ove uto tegrle 42 Alto to Hermte-Hmr s Iequlty g g g g g g The the equlty : rove tht the utos g g re - Lemm 43 Let e mesure o the tervl [ ] wth [ ] > 0 stsy the the oule equlty I oeets o the sum [ ]

7 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty 269 ] [ 22 hols or every ove uto ] [ : Proo The let-h se o the equlty 22 ollows rom Jese s equlty: ] [ ] [ The rght-h se o the equlty 22 ollows rom the hor equlty 6: l ho } { the ] [ ] [ } { l ho euse ho } { ho } { The equlty 22 ws stte or ove otuous utos rel Borel mesures [] lso or otuous mesures [9 Corollry 38] Remr 44 The oeets o the equlty 22 re uque e lulte y the ormuls 5 I ] [ / the ltertve resetto o the equlty 22 res s ollows: ] [ 23 The equlty 22 e geerlze y lto o the ormul 8: Theorem 45 Let e mesure o the tervl ] [ wth 0 > ] [ Let [0] ] [ e oeets o the sum Let ] [ : e ove uto I the equlty 2 s vl the the oule equlty ] [ 24 hols or 0 the oule equlty ] [ 25 hols or 0 Proo Puttg ] [ / we hve the reresetto

8 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty 270 Suose 0 Comg the lto o the ormul 8 wth the ove equlty t ollows ] [ Alyg the tegrl orm o Jees s equlty o the lst memer o the ove equlty the the rghth se o the equlty 22 we get ] [ ] [ ter whh rses the equlty 24 Suose 0 Usg the srete orm o Jese s equlty t ollows Ater usg the rght-h se o the equlty 22 multle y we ot ] [ esurg the equlty APPLICATION TO QUASI-ARITHMETIC MEANS Let e mesure o set wth 0 > : e strtly mootoe otuous uto tht s -tegrle o The -qus-ryeter o the set wth reset to the mesure e ee s the ot M 26 I s tervl the M elogs to euse / elogs to The ormul 26 lso e le to the -qus-ryeter eto o o-egtve or o-ostve -tegrle uto : wth 0 S M 27 ssumg -tegrlty o the uto Alyg the e o â â the revous ormul we ee ; ] [ M 28 lug Corollry 42 get: Corollry 46 Let e mesure o the tervl ] [ Let ] [ : e ether o-egtve or oostve -tegrle uto wth 0 Let [0] ] [ e oeets o the

9 IJRRAS 7 3 Deemer 203 Pvć & Novosel O Jese s & Hermte-Hmr s Iequlty sum Let :[ ] e strtly mootoe otuous utos I s ether -ove resg or -ove eresg the the equlty hols rove tht the utos M [ ] ; M [ ] ; 29 re -tegrle I s ether -ove eresg or -ove resg the the reverse equlty s vl 29 Proo Let us rove the se whe s -ove resg Usg the equlty 20 wth the ove uto the mootoe uto g whh se g we get The equlty 29 ollows ter lyg the resg uto o the ove equlty More o geerl orms reemets o the qus-rthmet mes e ou [6] 5 REFERENCES [] A M F "A est ossle Hmr equlty" Mthemtl Iequltes & Altos vol [2] J Hmr "Étue sur les rorétés es otos etères et e rtuler ue oto oserée r Rem" Jourl e Mthémtques Pures et Alquées vol [3] C Hermte "Sur eu lmtes ue tégrle ée" Mthess vol [4] J L W V Jese "Sur les otos ovees et les égltés etre les vleurs moyees" At Mthemt vol [5] A MD Merer "A vrt o Jese s equlty" Jourl o Iequltes Pure Ale Mthemts vol 4 o 4 Artle [6] J M Z Pv J Pe r "The equltes or qusrthmet mes" Astrt Ale Alyss vol 202 Artle ID ges 202 [7] D S Mtrov I B Lov "Hermte ovety" Aequtoes Mthemte vol [8] C P Nulesu L E Persso Cove Futos Ther Altos C Mthemtl Soety Srger New Yor USA 2006 [9] Z Pv "Cove omtos ryeters ove utos" Jourl o Iequltes Altos vol 203 Artle 6 3 ges 203 [0] Z Pv "Geerlztos o Jese s equlty" sumtte 0 ges 203 [] W Ru Rel Comle Alyss MGrw-Hll New Yor USA