Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 htt://wwwjournlofineulitiesndlictionscom/content/3//333 R E S E A R C H Oen Access On some ineulities for s-convex functions nd lictions Muhmet Emin Özdemir, Çetin Yıldız *, Ahmet Ock Akdemir nd Erhn Set 3 * Corresondence: yildizc@tuniedutr Dertment of Mthemtics, KK Eduction Fculty, Atturk University, Erzurum, 54, Turkey Full list of uthor informtion is vilble t the end of the rticle Abstrct Some new results relted to the left-hnd side of the Hermite-Hdmrd tye ineulities for the clss of functions whose second derivtives t certin owers re s-convex functions in the second sense re obtined Also, some lictions to secil mens of rel numbers re rovided MSC: Primry 6A5; 6D5 Keywords: Hdmrd s ineulity; s-convex concve function; Hölder ineulity; ower-men ineulity Introduction The following definition is well known in the literture: function f : I R, I R,is sid to be convex on I if the ineulity f tx ty tf x tf y holds for ll x, y I nd t [, ] Geometriclly, this mens tht if P, Q nd R re three distinct oints on the grh of f with Q between P nd R, then Q is on or below the chord PR In their er [], Hudzik nd Mligrnd considered, mong others, the clss of functions which re s-convexin thesecond sense This clssis defined in the followingwy: function f :[, R is sid to be s-convex in the second sense if f tx ty t s f x t s f y holds for ll x, y [,, t [, ] nd for some fixed s, ] The clss of s-convex functions in the second sense is usully denoted by Ks Itcnbeesilyseenthtfors =s-convexity reduces to the ordinry convexity of functions defined on [, In the sme er [], Hudzik nd Mligrnd roved tht if s,, f Ks imlies f [, [,, ie, they roved tht ll functions from Ks, s,, re nonnegtive Exmle [] Lets, nd, b, c RWedefinethefunctionf :[, R s, t =, f t= bt s c, t > 3 Özdemir et l; licensee Sringer This is n Oen Access rticle distributed under the terms of the Cretive Commons Attribution License htt://cretivecommonsorg/licenses/by/, which ermits unrestricted use, distribution, nd reroduction in ny medium, rovided the originl work is roerly cited
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge of htt://wwwjournlofineulitiesndlictionscom/content/3//333 It cn be esily checked tht i if b nd c,thenf Ks, ii if b >nd c <,thenf / Ks Mny imortnt ineulities re estblished for the clss of convex functions, but one of the most fmous is the so-clled Hermite-Hdmrd s ineulity or Hdmrd s ineulity This double ineulity is stted s follows: Let f be convex function on [, b] R, where bthen b f b f fb For severl recent results concerning Hdmrd s ineulity, we refer the interested reder to [ 5] In [6] Drgomir nd Fitztrick roved vrint of Hdmrd s ineulity which holds for s-convex functions in the second sense Theorem Suose tht f :[, [, is n s-convex function in the second sense, where s,, nd let, b [,, < b If f L[, b], then the following ineulities hold: b s f b f fb s The constnt k = /s is best ossible in the second ineulity in [7] The bove ineulities re shr For recent results nd generliztions concerning s-convex functions, see [8 3] Along this er, we consider rel intervl I R, nd we denote tht I is the interior of I The min im of this er is to estblish new ineulities of Hermite-Hdmrd tye for the clss of functions whose second derivtives t certin owers re s-convex functions in the second sense Min results To rove our min results, we consider the following lemm Lemm Let f : I R R be differentible ming on I where, b Iwith< b If f L[, b], then the following eulity holds: b f [ b = b t f t b t t f tb t b ]
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 3 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 Proof By integrtion by rts, we hve the following identity: I = t f t b = t b f = b f t b b f b = b b f 8 b t t 4 b [ t b b f t b ] t b 4 t b 8 b f f t b t tf t b t t Using the chnge of the vrible x = t b t for t [, ] nd multilying the both sides by b,weobtin b = b 8 f t f t b t b Similrly, we observe tht b = b 8 f b f b 3 b t f tb t b b f b b b 4 Thus, dding 3nd4, we get the reuired identity Theorem Let f : I [, R be differentible ming on I such tht f L[, b], where, b Iwith< b If f is s-convex on [, b], for some fixed s, ], then the following ineulity holds: b f b b 8s s s 3 { f s s b f f b 5 [ s s ]b { f f b 6 8s s s 3
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 4 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 Proof From Lemm,wehve b f b [ b t f t b t t f tb t b ] b [ b t t s f t s f ] [ b t t s f b ] t s b f [ b = b s 3 f f ] s s s 3 [ b f b ] b s s s 3 s 3 f b { f = b s s f 8s s s 3 wherewehveusehefcttht f b, 7 t t s = t s = t t s = s s s 3, t s = s 3 This roves ineulity 5 To rove 6, nd since f is s-convex on [, b], for ny t [, ], then by wehve b s f Combining 7nd8, we hve f fb 8 s b f b b { f s s b 8s s s 3 f f b b { f s f fb s s f b 8s s s 3 s = [ s s ]b { f f b, 8s s s 3 which roves ineulity 6, nd thus the roofis comleted
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 5 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 Corollry In Theorem, if we choose s =,we hve b f b b 9 { f 6 b f f b b { f f b 9 48 The next theorem gives new uer bound of the left Hdmrd ineulity for s-convex mings Theorem 3 Let f : I [, R be differentible ming on I such tht f L[, b], where, b Iwith< b If f is s-convex on [, b], for some fixed s, ] nd >with =,then the following ineulity holds: b f b b s [ f b b f f f b ] Proof Suose tht > FromLemm nd using the Hölder ineulity, we hve b f b [ b t f t b t t f tb t b ] b t f t b t b t f tb t b Becuse f is s-convex, we hve f t b t s { f b f nd f tb t b s { f b f b
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 6 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 By simle comuttion, nd t = t = t = Therefore, we hve b f b b s [ f b b f f f b ] This comletes the roof Corollry Let f : I [, R be differentible ming on I such tht f L[, b], where, b Iwith< b If f is s-convex on [, b], for some fixed s, ] nd >with =,then the following ineulity holds: b f b b s { s s s [ f f b ] Proof We consider ineulity, nd since f is s-convex on [, b], then by we hve b s f Therefore b f b b f fb s s [{ s s f s f b s f { s s f b ]
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 7 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 We let = s s f, b = s f b, = s f nd b = s s f b Here, < / <for >Usingthefct n i b i r i= n n r i i= i= b r i for < r <,,,, n ndb, b,,b n, we obtin the ineulities b f b b s [{ s s f s f b s f { s s f b ] b s { s s s [ f f b ] Theorem 4 Let f : I [, R be differentible ming on I such tht f L[, b], where, b Iwith< b If f, is s-convex on [, b], for some fixed s, ], then the following ineulity holds: b f b b { f b 3 s s s 3 s 3 f b s 3 f f b s s s 3 Proof Suose tht From Lemm nd using the ower men ineulity, we hve b f b [ b t f t b t t f tb t b ] b t t f t b t b t t f tb t b
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 8 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 Becuse f is s-convex, we hve t f t b t f b s s s 3 s 3 f nd t f tb t b b s 3 f f b s s s 3 Therefore, we hve b f b b { f b 3 s s s 3 s 3 f b s 3 f f b s s s 3 Corollry 3 In Theorem 4, if we choose s =,we hve b f b b 3 { f b 48 4 3 f b f f b 3 Now, we give the followinghdmrd-tye ineulityfor s-concve mings Theorem 5 Let f : I [, R be differentible ming on I such tht f L[, b], where, b Iwith< b If f is s-concve on [, b], for some fixed s, ] nd >with =,then the following ineulity holds: b f b b s / { 3 b f 3b 4 f 4 Proof From Lemm nd using the Hölder ineulity for >nd =,weobtin b f b b [ t f t b t t f tb t b ]
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge 9 of htt://wwwjournlofineulitiesndlictionscom/content/3//333 b t f t b b t t f tb t b Since f is s-concve, using ineulity, we hve f t b 3 b t s f 4 3 nd f tb t b 3b s f 4 4 From -4, we get b f b b s / { 3 b f 3b 4 f 4, which comletes the roof Corollry 4 In Theorem 5, if we choose s =nd 3 < <, >,we hve b f b b { 3 b f 3b 4 f 4 5 3 Alictions to secil mens We now consider the mens for rbitrry rel numbers α, β α β We tke: Arithmetic men: Aα, β= α β, α, β R ; Logrithmic men: Lα, β= Generlized log-men: α β ln α ln β, α β, α, β,α, β R ; [ β n α n ] n L n α, β=, n Z\{,, α, β R n β α Now, using the results of Section, we give some lictions to secil mens of rel numbers
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge of htt://wwwjournlofineulitiesndlictionscom/content/3//333 Proosition Let < < bnds, Then we hve A s, b L s s, b ss b { s 6 9 b s b s Proof The ssertion follows from 9 lied to the s-convexfunctioninthesecond sense f :[,] [, ], f x=x s Proosition Let < < bnds, Then we hve 3 {[ s 4 3 A s, b L s s, b ss b 48 [ b s bs 3 ] b s ] Proof The ssertion follows from lied to the s-convex function in the second sense f :[,] [, ], f x=x s Proosition 3 Let < < b nd > Then we hve { A s, b L s s, b b 3 b 3b Proof The ineulity follows from 5 lied to the concve function in the second sense f :[; b] R, f x=ln x The detils re omitted Cometing interests The uthors declre tht they hve no cometing interests Authors contributions ÇY, AOA nd ES crried out the design of the study nd erformed the nlysis MEÖ rticited in its design nd coordintion All uthors red nd roved the finl mnuscrit Author detils Dertment of Mthemtics, KK Eduction Fculty, Atturk University, Erzurum, 54, Turkey Dertment of Mthemtics, Fculty of Science nd Letters, Ağrı İbrhim Çeçen University, Ağrı, 4, Turkey 3 Dertment of Mthemtics, Fculty of Science nd Arts, Ordu University, Ordu, Turkey Received: 9 November Acceted: 4 July 3 Published: July 3 References Hudzik, H, Mligrnd, L: Some remrks on s-convexfunctions Aeu Mth 48, - 994 Özdemir, ME, Kırmcı, US: Two new theorems on mings uniformly continuous nd convex with lictions to udrture rules nd mens Al Mth Comut 43, 69-74 3 3 Drgomir, SS, Perce, CEM: Selected Toics on Hermite-Hdmrd Ineulities nd Alictions RGMIA Monogrhs Victori University, Melbourne htt://jmorg/rgmia/monogrhsh 4 Set, E, Özdemir, ME, Drgomir, SS: On the Hermite-Hdmrd ineulity nd other integrl ineulities involving two functions J Ineul Al, Article ID 48 5 Drgomir, SS, Agrwl, RP: Two ineulities for differentible mings nd lictions to secil mens of rel numbers nd to trezoidl formul Al Mth Lett 5,9-95 998 6 Drgomir, SS, Fitztrick, S: The Hdmrd s ineulity for s-convex functions in the second sense Demonstr Mth 34, 687-696 999 7 Pečrić, JE, Proschn, F, Tong, YL: Convex Functions, Prtil Orderings, nd Sttisticl Alictions, 37 Acdemic Press, Boston 99 8 Kırmcı, US, Bkul, MK, Özdemir, ME, Pečrić, J: Hdmrd-tye ineulities for s-convex functions Al Mth Comut 93, 6-35 7 9 Alomri, M, Drus, M: Hdmrd-tye ineulities for s-convex functions Int Mth Forum 34, 965-975 8
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 Pge of htt://wwwjournlofineulitiesndlictionscom/content/3//333 Alomri, M, Drus, M: The Hdmrd s ineulity for s-convex functions of -vribles Int J Mth Anl 3, 69-638 8 Hussin, S, Bhtti, MI, Ibl, M: Hdmrd-tye ineulities for s-convex functions Punjb Univ J Mth 4, 5-6 9 Avci, M, Kvurmci, H, Özdemir, ME: New ineulities of Hermite-Hdmrd tye vi s-convex functions in the second sense with lictions Al Mth Comut 7,57-576 3 Sriky, MZ, Set, E, Özdemir, ME: On new ineulities of Simson s tye for s-convex functions Comut Mth Al 6, 9-99 doi:86/9-4x-3-333 Cite this rticle s: Özdemir et l: On some ineulities for s-convexfunctions nd lictions Journl of Ineulities nd Alictions 3 3:333