Integral Form of Popoviciu Inequality for Convex Function
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1 Procees of e Paksa Acaey of Sceces: A. Pyscal a ozaoal Sceces 53 3: oyr Paksa Acaey of Sceces ISSN: r ole Paksa Acaey of Sceces Researc Arcle Ieral For of Pooc Ieqaly for oe Fco Kra Al Ka Tasaq Naz * a Jos Pečarć 2 Deare of aeacs Uersy of Saroa Saroa 4000 Paksa 2 Facly of Tele Tecoloy Uersy of Zareb 0000 Zareb roaa Absrac: I s aer e ew eral for of Pooc eqaly for coe fcos s cosrce a also e ew refee of eral for of Jese s eqaly s e. For e rose of alcao soe ew qas arec eas are efe alo w er ooocy roery. Keywors: oe fco Pooc eqaly Jese s eqaly qas arec eas. INTRODUTION AND PRELIINARY RESULTS A fco for all : R were s a coe sbse of real ecor sace s sa o be coe f a by a b y y a a b 0 sc a a b see [0 ae ~]. I [0 ae ~43] e Jese s eqaly scree erso s e as follows: Teore. Le be a coe sbse of real ecor sace 0] sc a a c c e : R be coe fco c c. 2 I [0 ae ~63] e eral for of Jese s eqaly s efe as follows. Teore.2 Le be a erable fco o a robably sace A ak ales a eral I R. If s a coe fco o I sc a e cooso fco s erable e. I [2] Bre c Pearce a Pe c arc e e refee of eral for of Jese s eqaly 3 by s refee of scree Jese s eqaly. oreoer [7] László orá a Pečarć e Recee ay 206; Accee As 206 *orreso aor: Kra Al Ka; Eal krass@al.co 3
2 340 Kra Al Ka e al e roee of eral for of Jese s eqaly 3 by s soe refee of scree Jese s eqaly wc s eeralzao of resl e [2] ey also e ew qas arec eas a roe er ooocy. Te Pooc eqaly s e by see [0 ae 73]. Teore.3 Le N sc a 3 2 [ a b] R be a eral [ a b] be a -le sc a 0 2 w. Also le :[ a b] R be a coe fco. Te were a 4 :!!!. I e crre cery e Pooc eqaly 4 s se by ay aors. I e oora [6] e eeralzao of 4 for real wes e syerc eas eoeal coey ea ale eores a acy eas are se. I [8 9] e eral erso a refee of a secal case of 4 s roe resecely. I [] e er eso aaloe of a secal case of 4 s e. oreoer [3 4 5] 4 s eeralze for er orer coe fcos a ffere erola olyoals. We se e ea of Bre c Pearce a Pe c arc e for Jese s eqaly [2] o cosrc e eral for of Pooc eqaly 4. Also follow e way of László orá a J. Pe c ar c e for refees of Jese s eqaly [7] we e alcao o e qas arec eas AIN RESULTS We ow coser soe yoeses wc are se or work.. Le E be a robably sace a le be ose bers w 2 Le : I R be a erable fco. 3 Le be a coe fco o eral I sc a e cooso s erable.
3 Ieral For of Pooc Ieqaly for oe Fco 34 Le 2 be a fe eer. Te σ -alebra k eerae by e roeco a : l r k l l l r : 6 s eoe by k E. A s efe as e roc easre o E s easre s qely s σ -fe secfe by :. l B B B B B l E 7 Teore 2. Asse - 3 e e follow eqales ol. a.. b.. Proof. a O era e eqaly 4 oer a relac by we ae
4 342 Kra Al Ka e al. O slfcao we ae. Ts es. b Us e scree Jese s eqaly e las er of eqaly e a a o sol we ae s es
5 Ieral For of Pooc Ieqaly for oe Fco 343. Uer e yoess 2 a 3 efe e fco o [0] e by. 8 Teore 2.2 For 2 be a eer we asse - 3 a coser :[0] RR as efe 8 e e follow saees are al. a. s coe. b. 0 [0] c. a [0]. s creas. Proof. a Sose [0] w a [0] e fro 8 we ae. O slfcao we ae
6 344 Kra Al Ka e al. By coey of we ae a s. Terefore s coe fco. b By e eral fro Jese s eqaly 8 yels or I 9 were
7 Ieral For of Pooc Ieqaly for oe Fco 345 I so fro 9 we ae 0 [0]. c. 0 0 [0]. Sce s coe a 0 [0] erefore for so Teore 2.3 Asse 2 a 3 0 e. 0 2 we ae Proof. Us b a c of Teore 2.2 we e frs wo eqales a for e las eqaly 0
8 346 Kra Al Ka e al Us e scree Jese s eqaly we ae s es. Reark 2.4 A refee slar o 0 of eral for of Jese s eqaly s roe Prooso 7 of [7].. 3. NEW QUASI-ARITETI EANS Now we roce soe ew qas arec eas. For s frs asse soe coos: 4 Le : I were I R be a eral s easrable. 5 Le : I R are coos a srcly oooe fcos. Defo Asse a 5. 4 For [0] we efe e class of qas-arec ea e by : were e erals are sose o be es. cooso Asse 6 le η : I R be a coos a srcly oooe fco sc a e η s erable o. Defe e ea η η η. 2
9 Ieral For of Pooc Ieqaly for oe Fco 347 Teore 3. Asse 4 5 a asse a a a erable o. a If s coe w s creas or s cocae w s ecreas e 3 ols for all [0]. b If s coe w s ecreas or s cocae w s creas e 4 ols for all [0]. Proof. a Us ar of fcos a I s a eral Teore 2.3 we ae. Us e scree Jese eqaly o e r se of las eqaly we e. O ak o bo ses we ae 3. b Slarly s e ar of fcos a Teore 2.3 were s cocae. O ak e we ae 4.
10 348 Kra Al Ka e al 4. REFERENES. Becze..P. Nclesc & F. Pooc. Poocs eqaly for fcos of seeral arables. Joral of aeacal Aalyss a Alcaos 365: Bre c I..E.. Pearce & J. Pečarć. Refees of Jese s eqaly. Taka Joral of aeacs 3: B S.I. K.A.Ka & J. Pečarć. Geeralzao Of Pooc Ieqaly For er Orer oe Fcos Va Taylor Polyoal. Aca Uersas Aless 42: BS. I.K. A.Ka & J.Pe c arc. Pooc ye eqales a Gree fco a eeralze ooery ey. aeacal Ieqales a Alcaos 8: BS. I. K. A. Ka & J. Pečarć. Pooc ye eqales a Gree fco a Taylor olyoal. Trks Joral of aeacs 40: orál. K.A. Ka & J. Pečarć. obaoral Iroees of Jeses Ieqaly / lasscal a New Refees of Jeses Ieqaly w Alcaos. ooras Ieqales 8 Zareb: Elee orá L. & J. Pečarć. Refee of e classcal Jese s eqaly co fro refee of e scree Jese s eqaly. Aaces Ieqales a Alcaos 33: Nclesc. P. Te eral erso of Pooc s eqaly. Joral of aeacal Ieqales 33: Nclesc.P. & F. Pooc. A refee of Poocs eqaly. Blle of e Socey for aeacal Sceces Roaa 49: Pečarć J. F. Prosca & Y.L. To. oe fcos Paral Orers a Sascal Alcaos Acaec Press New York 992.
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