On Some Hadamard-Type Inequalıtıes for Convex Functıons

Transcription

1 Aville t htt://vuedu/ Al Al Mth ISSN: Vol 9, Issue June 4, Alictions nd Alied Mthetics: An Intentionl Jounl AAM On Soe Hdd-Tye Inequlıtıes o, Convex Functıons M Ein Özdei Detent o Mthetics Attuk Univesity 564, Kus, Ezuu, Tukey eos@tuniedut Ehn Set Detent o Mthetics Odu Univesity Odu, Tukey ehnset@yhooco Ahet Ock Akdei Detent o Mthetics Ağı İhi Çeçen Univesity 4, Ağ, Tukey hetkdei@giedut Received: August 8, 3; Acceted: August, 3 Astct In this e, we deine new clss o convex unctions which is clled, convex unctions We lso ove soe Hdd's tye inequlities sed on this new deinition Keywods: convex; Hdd's inequlity; convex,, convex MSC No: 6D5, 6A7 Intoduction The ollowing deinition is well known in the litetue: unction : I R, I R, is sid to e convex on I i the inequlity 388

2 AAM: Inten J, Vol 9, Issue June tx ty t x t y holds o ll x y I t, Geoeticlly, this ens tht i P, Q nd R e thee distinct oints on the gh o with Q etween P nd R, then Q is on o elow chod PR Let : I R R e convex unction deined on the intevl I o el nues nd, I with Then the ollowing doule inequlity holds o convex unctions:, nd x This inequlity is well known in the litetue s Hdd's inequlity Pece et l 998 genelized this inequlity to convex ositive unction which is deined on n intevl [, ], o ll x, y [, ] nd [,]; x y x y, i, x y, i, Clely convex unctions e sily log convex unctions nd convex unctions e odiny convex unctions Anothe inequlity which is well known in the litetue s Minkowski Inequlity is stted s ollows; Let, x, nd g x Then, x g x x g x Deinition A unction : I, o, equivlently, i o ll x, y I nd, is sid to e log-convex o ultilictively convex i log is convex, t one hs the inequlity: t tx t y x y t, [Pečić et l 99]

3 39 M Ein Özdei et l We note tht log convex unction is convex, ut the convese y not necessily e tue Ngoc et l 9 estlished ollowing theoes o convex unctions: Theoe Let :[, ], e convex unction on [, ] with Then the ollowing inequlity holds o : x 3 Theoe Let, g :[, ], e convex nd s convex unctions esectively on [, ] with Then, the ollowing inequlity holds o, x g x s s s s g g s s 4 Theoe 3 Let, g :[, ], e convex nd s convex unctions esectively on [, ] with Then the ollowing inequlity holds i, nd : s s g g s s x g x 5 Gill et l 997 oved the ollowing inequlity o convex unctions Theoe 4 Suose is ositive convex unction on [, ] Then, tdt L, 6

4 AAM: Inten J, Vol 9, Issue June 4 39 I is ositive concve unction, then the inequlity is evesed Fo elted esults on convexity see [Yng nd Hwng, Gill et l 997 nd Ngoc et l 9] Tode 985 deined convex unctions, s ollows: Deinition The unction :, R,, is sid to e convex, whee,, i we hve tx t y t x t y,, nd t, We sy tht is concve i is convex o ll x y, We ee to the es [Bkul et l 6; Bkul et l 7; Özdei et l nd Tode 988] involving inequlities o convex unctions Dgoi nd Tode 993 oved the ollowing inequlity o convex unctions Theoe 5 Let :, R e convex unction with,] I nd L,, then one hs the inequlity:, in x Dgoi oved soe Hdd-tye inequlities o 7 convex unctions s ollows Theoe 6 Let :, R e convex unction with,] I nd L,, then one hs the inequlity: Theoe 7 x 4 x 8 Let :, R e convex unction with,], then one hs the inequlity: I L, whee

5 39 M Ein Özdei et l x x 9 Min Results We will stt with the ollowing deinition Deinition 3 A ositive unction is, convex on,, i o ll, y,,, x y x y, i x y, i x, nd This deinition o, convexity ntully coleents the concet o, concvity in which the inequlity is evesed Rek We hve tht, convex unctions e sily log convex unctions nd, convex,, unctions e odiny convex unctions on Rek We hve tht, convex unctions e convex unctions Rek 3 We hve tht, convex unctions e convex unctions Now, we will ove soe inequlities sed on ove deinition nd eks Theoe 8 Suose tht is, convex unction on,, Then, we hve the inequlity; tdt L,, o I is, concve unction, then the inequlity is evesed

6 AAM: Inten J, Vol 9, Issue June Poo : Let, Fist ssue tht By the deinition o, convexity, we cn wite t dt s s s s ds ds Using the ct tht L,, we otin the desied esult Siilly, o, we hve t dt s s ds s s ds L, Finlly, let, o, we hve t dt ds s s Couting the ight hnd side o the ove inequlity, we get t dt L, The oo o the othe cse such s, y e otined in siil wy

7 394 M Ein Özdei et l Rek 4 In Theoe 8, i we choose, we hve the inequlity 6 Theoe 9 Let :,,, e ollowing inequlity holds:, convex unction on, with Then, the x, o Poo: Since is, convex unction nd, we cn wite t t t t o ll t,, It is esy to oseve tht x Using the inequlity, we get t t dt t t dt t t dt t dt t Thus, dt

8 AAM: Inten J, Vol 9, Issue June x, which coletes the oo Coolly In Theoe 9, i we choose, ollowing inequlity; convex unction on x, with Then, we hve the Coolly In Theoe 9, i we choose n, convex unction on, with Then, we hve the ollowing inequlity; x Rek 5 In Theoe 9, i we choose, the ight hnd side o Hdd's inequlity Theoe convex unction on, Let, g :,,, e, convex nd, Then, the ollowing inequlity holds;, with Then, we hve convex unction on, with x g x, g g o nd Poo: Since is, convex unction nd g is, convex unction, we hve t t t t

9 396 M Ein Özdei et l nd, gt t t g t g o ll t,, Since nd g e non-negtive uncions, hence t tgt t t t tg t g Integting oth sides o the ove inequlity ove, with esect to t, we otin t t g t t dt t t t g t g dt By lying Hölde's inequlity, we hve t t t g t g dt By using the ct tht g g t t dt t g t g dt x g x We otin the desied esult Coolly 3 t t g t t dt In Theoe, i we choose, nd x g x, we hve the ollowing inequlity; x

10 AAM: Inten J, Vol 9, Issue June Coolly 4 In Theoe, i we choose nd, we hve the ollowing inequlity; x g x g g Theoe Let :,,, e, with,] I nd L [, ], then one hs the ollowing inequlity; Poo:, convex unction on in,,, x L L Since is, convex unction, we cn wite t x t y tx t y o ll x, y nd, which gives nd t t t t 3 t t t t, 4 o, t Integting oth sides o 3 ove, with esect to t, we otin o t t dt t t dt,

11 398 M Ein Özdei et l dt t t x Now, suose tht, Fist ssue tht Then, we get, L x Siilly, o, we hve, L dt t t x Finlly, let, o, we hve gin, L dt t t x When, the oo is siil So, we otin the inequlity, L x Anlogously, y integting oth sides o the inequlity 4, we otin,, x L which coletes the oo Rek 6 In Theoe, i we choose, we hve the inequlity 7

12 AAM: Inten J, Vol 9, Issue June Rek 7 In Theoe, i we choose, we hve the inequlity 6 Rek 8 In Theoe, i we choose, we hve the ight hnd side o Hdd's inequlity Theoe Let :,,, e L [, ], then one hs the ollowing inequlities;, convex unction on, with,] I Poo: x 4 x 5 By the, convexity o, we hve tht x y y x, o ll x, y, I we tke x t t nd y t t, we deduce t t t t, o ll, t Integting the esult ove, with esect to t, we get t t dt t t dt 6 Tking into ccount tht t t dt x nd

13 4 M Ein Özdei et l t t dt x in 6, we otin the ist inequlity o 5 By the, convexity o, we lso hve tht t t t t o ll, t t t t, t Integting the ove inequlity ove, with esect to t, we get x x By siil guent, we cn stte x x 8, which coletes the oo Rek 9 In theoe, i we choose, we hve the inequlity 8 Rek In theoe, i we choose, we hve the Hdd's inequlity Conclusion In this e, new clss o convex unctions clled, convex unctions hve een deined nd soe new Hdd-tye inequlities hve een otined

14 AAM: Inten J, Vol 9, Issue June 4 4 Acknowledgeent The uthos would like to thnk the eeees o thei encouging ttitude nd vlule suggestions in the eview ocess o the wok REFERENCES Bkul, MK, Pečić, J nd Riičić, M 6 Conion inequlities to Jensen's inequlity o convex nd, convex unctions, J Ineq Pue nd Al Mth, 7 5, At 94 Bkul, MK, Özdei, ME nd Pečić, J 7 Hdd-tye inequlities o convex nd, convex unctions, J Inequl Pue nd Al Mth, 9, 4, Aticle 96 Dgoi, SS On soe new inequlities o Heite-Hdd tye o convex unctions, Tkng Jounl o Mthetics, 33 Dgoi, SS nd Tode, G 993 Soe inequlities o convex unctions, Studi Univ Beş-Bolyi Mth, 38, -8 Gill, PM, Pece, CEM nd Pečić, J 997 Hdd's inequlity o convex unctions, Jounl o Mth Anlysis nd Al, 5, Ngoc, NPG, Vinh, NV nd Hien, PTT 9 Integl inequlities o Hdd-tye o convex unctions, Intentionl Mtheticl Fou, 4, Özdei, ME, Avcı, M nd Set, E On soe inequlities o Heite-Hdd tye vi convexity, Alied Mthetics Lettes, 3, 65-7 Pece, CEM, Pečić, J, Siić, V 998 Stolsky Mens nd Hdd's Inequlity, Jounl Mth Anlysis Al,, 99-9 Pečić, J, Poschn, F nd Tong, YL 99 Convex Functions, Ptil Odeings nd Sttisticl Alictions, Acdeic Pess, Inc Tode, G 985 Soe geneliztion o the convexity, Poc Colloq Aox Ot, Cluj- Noc, Tode, G 988 On geneliztion o the convexity, Mthetic, 3 53, Yng, GS nd Hwng, DY Reineents o Hdd's inequlity o convex unctions, Indin Jounl Pue Al Mth, 3,