f(tx + (1 t)y) h(t)f(x) + h(1 t)f(y) (1.1)

Transcription

1 MATEMATIQKI VESNIK 68, 206, Mrch 206 origili uqi rd reserch pper INTEGRAL INEQUALITIES OF JENSEN TYPE FOR λ-convex FUNCTIONS S. S. Drgomir Abstrct. Some itegrl iequlities o Jese type or λ-covex uctios deied o rel itervls re give.. Itroductio Assume tht I d J re itervls i R, 0, J d uctios h d re rel o-egtive uctios deied i J d I, respectively. Deiitio. 20 Let h: J 0, ith h ot ideticl to 0. We sy tht : I 0, is h-covex uctio i or ll x, y I e hve or ll t 0,. tx + ty htx + h ty. For some results cocerig this clss o uctios see 3, 3, This clss o uctios cotis the clss o Goduov-Levi type uctios 9, 0, 4, 6. It lso cotis the clss o P uctios d qusi-covex uctios. For some results o P -uctios see 5 hile or qusi covex uctios, the reder c cosult. Deiitio 2. 4 Let s be rel umber, s 0,. A uctio : 0, 0, is sid to be s-covex i the secod sese or Brecker s-covex i or ll x, y 0, d t 0,. tx + ty t s x + t s y For some properties o this clss o uctios see, 2, 4, 7, 8, Mthemtics Subject Clssiictio: 26D5, 25D0 Keyords d phrses: Covex uctio; discrete iequlity; λ-covex uctio; Jese type iequlity. 45

2 46 S. S. Drgomir We c itroduce o other clss o uctios deied o covex subset C o lier spce X tht cotis s limitig cses the clsses o Goduov-Levi d P -uctios. Deiitio 3. We sy tht the uctio : C X 0, is o s-goduov- Levi type, ith s 0,, i or ll t 0, d x, y C. tx + ty t s x + y,.2 t s We observe tht or s = 0 e obti the clss o P -uctios hile or s = e obti the clss o Goduov-Levi. I e deote by Q s C the clss o s- Goduov-Levi uctios deied o C, the e obviously hve P C = Q 0 C Q s C Q s2 C Q C = QC or 0 s s 2. For dieret iequlities o Hermite-Hdmrd or Jese type relted to these clsses o uctios, see, 3, 3, 5 9. A uctio h: J R is sid to be supermultiplictive i hts hths or y t, s J..3 I the iequlity.3 is reversed, the h is sid to be submultiplictive. equlity holds i.3 the h is sid to be multiplictive uctio o J. I 5, e itroduced the olloig cocept o uctios: I the Deiitio 4. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0. A mppig : C R deied o covex subset C o lier spce X is clled λ-covex o C i αx + βy λαx + λβy.4 α + β λα + β or ll α, β 0 ith α + β > 0 d x, y C. We observe tht i : C R is λ-covex o C, the is h-covex o C ith ht = λt λ, t 0,. I : C 0, is h-covex uctio ith h supermultiplictive o 0,, the is λ-covex ith λ = h. We hve the olloig result providig my exmples o subdditive uctios λ: 0, 0,. Theorem. 5 Let hz = =0 z be poer series ith oegtive coeiciets 0 or ll N d coverget o the ope disk D0, R ith R > 0 or R =. I r 0, R the the uctio λ r : 0, 0, give by λ r t := l hr hr exp t is oegtive, icresig d subdditive o 0,..5

3 Itegrl iequlities o Jese type 47 No, i e tke hz = z, z D0,, the r exp t λ r t = l r is oegtive, icresig d subdditive o 0, or y r 0,. I e tke hz = expz, z C the.6 λ r t = r exp t.7 is oegtive, icresig d subdditive o 0, or y r > 0. Corollry. 5 Let hz = =0 z be poer series ith oegtive coeiciets 0 or ll N d coverget o the ope disk D0, R ith R > 0 or R = d r 0, R. For mppig : C R deied o covex subset C o lier spce X, the olloig sttemets re equivlet: i The uctio is λ r -covex ith λ r : 0, 0,, hr λ r t := l ; hr exp t ii We hve the iequlity αx+βy α+β hr hr exp α β or y α, β 0 ith α + β > 0 d x, y C. iii We hve the iequlity x y hr hr.8 hr exp α hr exp β hr exp α x hr exp β y αx+βy hr x+y αx+βy α+β.9 hr exp α β α+β or y α, β 0 ith α + β > 0 d x, y C. We observe tht, i the cse he λ r t = r exp t, t 0 the the uctio is λ r -covex o covex subset C o lier spce X i αx + βy exp αx + exp βy.0 α + β exp α β or y α, β 0 ith α + β > 0 d x, y C. Notice tht this deiitio is idepedet o r > 0. The iequlity.0 is equivlet to αx + βy expβexpα x + expαexpβ y. α + β expα + β or y α, β 0 ith α + β > 0 d x, y C. Motivted by the lrge iterest o Jese d Hermite-Hdmrd iequlities tht hs bee mterilized i the lst to decdes by the publictio o hudreds o ppers, e estblish here some iequlities o these types or λ-covex uctios deied o rel itervls.

4 48 S. S. Drgomir 2. Ueighted Jese itegrl iequlities The olloig discrete iequlity o Jese type hs bee obtied i 6: Theorem 2. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0 d mppig : C R deied o covex subset C o lier spce X. The olloig sttemets re equivlet: i is λ-covex o C; ii For ll x i C d p i 0 ith i {,..., }, 2 so tht P > 0, e hve the iequlity p i x i λp i x i. 2. P λp i= i= The proo c be doe by iductio over 2. Corollry 2. Let : C R be λ-covex uctio o C d α i 0,, i {,..., } ith i= α i =. The or y x i C ith i {,..., } e hve the iequlity α i x i λα i x i. 2.2 i= λ i= I prticulr, e hve x + + x c x + + x, 2.3 here c := λ λ. We hve the olloig versio o Jese s iequlity s ell: Corollry 3. Let : C R be λ-covex uctio o C d x i C d p i 0 ith i {,..., }, 2 so tht P > 0. The e hve the iequlity p i x i pi λ x i. 2.4 λ P i= i= P The proo ollos by 2.2 or α i = pi P, i {,..., }. We re ble o to stte d prove the olloig ueighted Jese iequlity or Riem itegrl: Theorem 3. Let u:, b m, M be Riem itegrble uctio o, b. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0

5 Itegrl iequlities o Jese type 49 d the uctio : m, M 0, is λ -covex d Riem itegrble o the itervl m, M. I the olloig limit exists the b λt lim = k 0, 2.5 t 0+ t k ut dt λb Proo. Cosider the sequece o divisios d the itermedite poits d : x i = + i b, i {0,..., } ξ i = + i b, i {0,..., }. ut dt. 2.6 We observe tht the orm o the divisio := mx i {0,..., } x 0 s d sice u is Riem itegrble o, b, the b ut dt = lim uξ i x i+ x i b = lim u i+ x i = + i b. Also, sice : m, M 0, is Riem itegrble, the u is Riem itegrble d b ut dt = lim u + i b. Utilisig the iequlity 2. or p i := b d x i := u + i b e hve b u + i b b b b λb b λ u + i b 2.7 or y. Observe tht lim λ b λt = lim = k 0,, b t 0+ t d by tkig the limit over i the iequlity 2.7, e deduce the desired result 2.6. Corollry 4. Let u:, b m, M be Riem itegrble uctio o, b d hz = =0 z be poer series ith oegtive coeiciets 0

6 50 S. S. Drgomir or ll N d coverget o the ope disk D0, R ith R > 0 or R = d r 0, R. Let λ r : 0, 0, be give by hr λ r t := l hr exp t d the uctio : m, M 0, be λ r -covex d Riem itegrble o the itervl m, M. The b rh r b ut dt ut dt. 2.8 b hr l hr hr exp b Proo. We observe tht λ r is dieretible o 0, d λ rt := r exp th r exp t hr exp t or t 0,, here h z = = z. Sice λ r 0 = 0, thereore λs k = lim = λ s 0+ s +0 = rh r > 0 or r 0, R. hr Utilisig 2.6 e get the desired result 2.8. The olloig Hermite-Hdmrd type iequlity holds: Corollry 5. With the ssumptios o Theorem 3 or d λ d i, b = m, M, e hve the Hermite-Hdmrd type iequlity + b k b t dt λb Remrk. Assume tht the uctio : m, M 0, is λ-covex d Riem itegrble o the itervl m, M ith λt = exp t, t 0. I u:, b m, M is Riem itegrble uctio o, b, the b b ut dt ut dt. b exp b I prticulr, or, b = m, M d ut = t e hve the Hermite-Hdmrd type iequlity + b b t dt. 2 exp b The proo ollos rom 2.6 observig tht λt k = lim = λ t 0+ t +0 =.

7 Itegrl iequlities o Jese type 5 Utilisig similr rgumet d the iequlity 2.4 e c stte the olloig result s ell: Theorem 4. Let u:, b m, M be Riem itegrble uctio o, b. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0 d the uctio : m, M 0, is λ -covex d Riem itegrble o the itervl m, M. I the limit 2.5 exists, the b k ut dt λb Exmples o such iequlities re icorported belo: ut dt. 2.0 Corollry 6. Let u:, b m, M be Riem itegrble uctio o, b d hz = =0 z be poer series ith oegtive coeiciets 0 or ll N d coverget o the ope disk D0, R ith R > 0 or R = d r 0, R. Let λ r : 0, 0, be give by hr λ r t := l hr exp t d the uctio : m, M 0, be λ r -covex d Riem itegrble o the itervl m, M. The b rh r b ut dt ut dt. 2. b b hr l hr hre We lso hve the Hermite-Hdmrd type iequlity: Corollry 7. With the ssumptios o Theorem 4 or d λ d i, b = m, M, e hve the Hermite-Hdmrd type iequlity + b k b t dt λb Remrk 2. Assume tht the uctio : m, M 0, is λ-covex d Riem itegrble o the itervl m, M ith λt = exp t, t 0. I u:, b m, M is Riem itegrble uctio o, b, the b ut dt e b e b ut dt. b I prticulr, or, b = m, M d ut = t e hve the Hermite-Hdmrd type iequlity + b e 2 e b t dt. b

8 52 S. S. Drgomir 3. Weighted Jese itegrl iequlities We c prove o eighted versio o Jese iequlity. Theorem 5. Let u, :, b m, M be Riem itegrble uctios o, b d t 0 or y t, b ith t dt > 0. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0 d the uctio : m, M 0, is λ-covex d Riem itegrble o the itervl m, M. I the olloig limit exists, is iite d t lim = l > 0, 3. t λt the b t dt tut dt Proo. Cosider the sequece o divisios d the itermedite poits b l t dt λtut dt. 3.2 d : x i = + i b, i {0,..., } ξ i = + i b, i {0,..., }. We observe tht the orm o the divisio := mx i {0,..., } x i+ x i = b 0 s. I e rite the iequlity 2. or the sequeces p i = + i b d x i = u + i b, i {0,..., } e get + i b λ + i b λ or. Observe tht + i b = b + i b b + i + i b u + i b u + i b, i b u + i b b + i b u + i b

9 d λ + i b λ Itegrl iequlities o Jese type 53 + i b u + i b = + i b λ + i b b + i b b The rom 3.3 e get b b λ + i b u + i b. + i b u + i b + i b + i b λ + i b b + i b b λ + i b u + i b or ll. Sice + i b lim = the t dt =, lim b = lim + i b λ + i + i b lim b = lim t λt = l d by lettig i 3.4 e get the desired result 3.2. The olloig ueighted versio o Jese iequlity holds: b 3.4 Corollry 8. Let u:, b m, M be Riem itegrble uctio o, b. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0 d the uctio : m, M 0, be λ -covex d Riem itegrble o the itervl m, M. I the limit 3. exists, the b b ut dt lλ ut dt. 3.5 b b Moreover, i, b = m, M, the by tkig ut = t, t, b, e hve the Hermite- Hdmrd iequlity + b b lλ t dt b

10 54 S. S. Drgomir Remrk 3. I order to give exmples o subdditive uctios λ: 0, 0, ith the property tht λt > 0 or ll t > 0 d or hich the olloig limit exists, is iite d lim t t = l > 0, 3.7 λt e cosider the poer series hz = = z ith oegtive coeiciets 0 or ll, > 0 d coverget o the ope disk D0, R ith R > 0 or R =. Let λ r : 0, 0, be give by λ r t := l hr hr exp t We ko tht λ r is dieretible o 0, d. λ rt = r exp th r exp t hr exp t or t 0,, here h z = = z. By l Hospitl s rule e hve lim t t λ r t = lim t λ rt. Sice or the poer series hz = z + 2 z z 3 + e hve h z = z z 2 +, the λ rt = r exp t r exp t r exp t 2 + r exp t + 2 r exp t + 3 r exp t 2 + = r exp t r exp t r exp t + 3 r exp t 2, t 0,. + Thereore lim t λ rt = d lim t t λ r t =. Corollry 9. Let u, :, b m, M be Riem itegrble uctios o, b d t 0 or y t, b ith t dt > 0. Cosider the poer series hz = = z ith oegtive coeiciets 0 or ll, > 0 d coverget o the ope disk D0, R ith R > 0 or R =. Let r 0, R d ssume tht the uctio : m, M 0, is λ r -covex d Riem itegrble o the itervl m, M ith hr λ r t := l. hr exp t The e hve the iequlity b b t dt tut dt t dt l hr hr exp t ut dt. 3.8

11 Itegrl iequlities o Jese type 55 The proo ollos by Theorem 5 observig tht l =. Remrk 4. With the ssumptios o Corollry 9 or u, h d e hve b hr b ut dt l b hre ut dt. 3.9 b I prticulr, or, b = m, M e hve the Hermite-Hdmrd iequlity + b hr b l 2 hre t dt. 3.0 b 4. Itervl depedecy Let u:, b m, M be Riem itegrble uctio o, b. Let λ: 0, 0, be uctio ith the property tht λt > 0 or ll t > 0 d the uctio : m, M 0, be λ-covex d Riem itegrble o the itervl m, M. Assume lso tht the olloig limit exists By Theorem 3 e hve tht, u, λ;, b := λt lim = k 0,. t 0+ t ut dt k λb b ut dt Theorem 6. With the bove ssumptios or u, λ, d k e hve: i For y c, b e hve, u, λ;, b, u, λ;, c +, u, λ; c, b 0, 4.2 i.e.,, u, λ; is superdditive uctio o itervls. ii For y c, d, b ith c < d e hve, u, λ;, b, u, λ; c, d 0, 4.3 i.e.,, u, λ; is mootoic odecresig uctio o itervls. Proo. i By the λ-covexity o e hve or c, b tht b ut dt b c c = ut dt + b c b b c b b c λc c c ut dt + λb c b c λb c c ut dt ut dt.

12 56 S. S. Drgomir Thereore, u, λ;, b = c c ut dt + ut dt + c λc c c c ut dt k λb b ut dt λb k =, u, λ;, c +, u, λ; c, b d the iequlity 4.2 is proved. ii Obvious by the property 4.2. ut dt + λb c b c λb c ut dt ut dt Remrk 5. I, b = m, M d ut = t, t, b the the uctiol δ, λ;, b := t dt + b k λb 0 2 is superdditive d mootoic odecresig uctio o itervls. REFERENCES M. Alomri d M. Drus, The Hdmrd s iequlity or s-covex uctio, It. J. Mth. Al. Ruse 2, 3 6, 2008, M. Alomri d M. Drus, Hdmrd-type iequlities or s-covex uctios, It. Mth. Forum 3, , M. Bombrdelli d S. Vrošec, Properties o h-covex uctios relted to the Hermite- Hdmrd-Fejér iequlities, Comput. Mth. Appl. 58, , W. W. Brecker, Stetigkeitsussge ür eie Klsse verllgemeierter kovexer Fuktioe i topologische liere Räume, Publ. Ist. Mth. Beogrd N.S , S. S. Drgomir, Iequlities o Hermite-Hdmrd type or λ-covex uctios o lier spces, Preprit RGMIA Res. Rep. Coll S. S. Drgomir, Discrete iequlities o Jese type or λ-covex uctios o lier spces, Preprit RGMIA Res. Rep. Coll S.S. Drgomir d S. Fitzptrick, The Hdmrd iequlities or s-covex uctios i the secod sese, Demostrtio Mth. 32, 4 999, S.S. Drgomir d S. Fitzptrick, The Jese iequlity or s-brecker covex uctios i lier spces, Demostrtio Mth. 33, 2000, S. S. Drgomir d B. Mod, O Hdmrd s iequlity or clss o uctios o Goduov d Levi, Idi J. Mth. 39, 997, 9. 0 S. S. Drgomir d C. E. M. Perce, O Jese s iequlity or clss o uctios o Goduov d Levi, Period. Mth. Hugr. 33, 2 996, S. S. Drgomir d C. E. M. Perce, Qusi-covex uctios d Hdmrd s iequlity, Bull. Austrl. Mth. Soc , H. Hudzik d L. Mligrd, Some remrks o s-covex uctios, Aequtioes Mth. 48, 994, M. A. Lti, O some iequlities or h-covex uctios, It. J. Mth. Al. Ruse 4, ,

13 Itegrl iequlities o Jese type 57 4 D. S. Mitriović d J. E. Pečrić, Note o clss o uctios o Goduov d Levi, C. R. Mth. Rep. Acd. Sci. Cd 2, 990, C. E. M. Perce d A. M. Rubiov, P-uctios, qusi-covex uctios, d Hdmrdtype iequlities, J. Mth. Al. Appl. 240, 999, M. Rdulescu, S. Rdulescu d P. Alexdrescu, O the Goduov-Levi-Schur clss o uctios, Mth. Iequl. Appl. 2, , M. Z. Sriky, A. Sglm, d H. Yildirim, O some Hdmrd-type iequlities or h- covex uctios, J. Mth. Iequl. 2, , M. Z. Sriky, E. Set d M. E. Özdemir, O some e iequlities o Hdmrd type ivolvig h-covex uctios, Act Mth. Uiv. Comei. N.S. 79, 2 200, M. Tuç, Ostroski-type iequlities vi h-covex uctios ith pplictios to specil mes, J. Iequl. Appl. 203: S. Vrošec, O h-covexity, J. Mth. Al. Appl. 326, 2007, received ; i revised orm ; vilble olie Mthemtics, College o Egieerig & Sciece, Victori Uiversity, PO Box 4428, Melboure City, MC 800, Austrli School o Computtiol & Applied Mthemtics, Uiversity o the Wittersrd, Privte Bg 3, Johesburg 2050, South Aric E-mil: sever.drgomir@vu.edu.u