New Jensen and Hermite Hadamard type inequalities for h-convex interval-valued functions

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1 Zho et l Journl o Inequlities nd Applictions 8 8:3 R E S E A R C H Open Access New Jensen nd Hermite Hdmrd type inequlities or h-convex intervl-vlued unctions Dng Zho,*, Tinqing An, Guoju Ye nd Wei Liu * Correspondence: dngzho@63com College o Science, Hohi University, Nnjing, Chin School o Mthemtics nd Sttistics, Hubei Norml University, Hungshi, Chin Abstrct In thispper, weintroduce theh-convex concept or intervl-vlued unctions By using the h-convex concept, we present new Jensen nd Hermite Hdmrd type inequlities or intervl-vlued unctions Our inequlities generlize some known results MSC: 6D5; 6E5; 8B Keywords: Intervl-vlued unctions; Jensen inequlity; Hermite Hdmrd inequlity; h-convex Introduction The theory o intervl nlysis hs long history which cn be trced bck to Archimedes computtion o the circumerence o circle It ell into oblivion or long time becuse o lck o pplictions to other sciences To the best o our knowledge, signiicnt work did not pper in this re until the 95s The irst monogrph on intervl nlysis is the celebrted book o RE Moore [8] One o the initil uses o intervl nlysis ws to compute the error bounds o the numericl solutions o inite stte mchine However, intervl nlysis hs emerged s very useul over the lst ity yers due to its mny pplictions in vrious ields We now see pplictions in utomtic error nlysis [4], computer grphics [45], neurl network output optimiztion [47], nd mny others For more undmentl results nd pplictions o intervl nlysis theory, we reer the reder to the ppers [7, 8,,, 37] nd monogrph [9] Recently, severl clssicl integrl inequlities hve been extended not only to the context o intervl-vlued unctions by Chlco-Cno et l [5, 6], Román-Flores et l [4, 4], Flores-Frnulič et l [9], Cost nd Román-Flores [],but lsotomoregenerlset- vlued mps by Mtkowski nd Nikodem [4], Mitroi et l [7], nd Nikodem et l [3] In prticulr, Cost [9] presented new uzzy version o Jensen inequlities type integrl or uzzy-intervl-vlued unctions Motivted by Cost [9] nd Drgomir [5], we introduce the h-convex concept or intervl-vlued unctions Under the h-convex concept, we present new Jensen type inequlities or intervl-vlued unctions The second objective o the rticle is to promote the ollowing inequlity which is known s the Hermite The Authors 8 This rticle is distributed under the terms o the Cretive Commons Attribution 4 Interntionl License which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthors nd the source, provide link to the Cretive Commons license, nd indicte i chnges were mde

2 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge o 4 Hdmrd inequlity [, ]: + b b x dx +b, where :[, b] R is convex unction For vrious interesting extensions nd generliztions o Hermite Hdmrd inequlities, see [6 8,, 6, 3, 34 36, 39, 48 5] In [43], Sriky et l proved vrint o the Hermite Hdmrd inequlity or h-convex unction s ollows Theorem Let :[, b] R be n h-convex unction nd h Then + b h b x dx [ +b ] ht dt Since, some urther reinements nd extensions o the Hermite Hdmrd inequlities or h-convex unctions hve been extensively studied in [, 4, 5, 3, 33, 44] Some Hermite Hdmrd nd Jensen type inequlities or strongly h-convex unctions were obtined lso in [, 3] In this pper, we estblish some Hermite Hdmrd type inequlities or h-convex intervl-vlued unctions Our results generlize the previous inequlities presented in [9, 3, 43] Thepperisorgnizedsollows Atersectionopreliminries, insect 3 the h-convex h-concve, h-ine concepts or intervl-vlued unctions re given Moreover, some Jensen type inequlities nd equlities re proved, respectively In Sect 4, we obtin some Hermite Hdmrd type inequlities or h-convex intervl-vlued unctions In Sect 5, we discuss the min results nd limittion o the present studies We end with Sect 6 o conclusions nd uture work Preliminries In this section, we recll some bsic deinitions, nottions, properties, nd results on intervl nlysis, which re used throughout the pper A rel intervl [u] is the bounded, closed subset o R deined by [u]=[u, u]={x R u x u}, where u, u R nd u uthenumbersu nd u re clled the let nd the right endpoints o [u, u], respectively When u nd u re equl, the intervl [u] is sid to be degenerte In this pper, the term intervl will men nonempty intervl We cll [u] positive i u >or negtive i u < The inclusion is deined by [u, u] [v, v] v u, u v For n rbitrry rel number λ nd [u], the intervl λ[u]isgivenby [λu, λu] iλ >, λ[u, u]= {} i λ =, [λu, λu] iλ <

3 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 3 o 4 For [u] =[u, u] nd[v] =[v, v], the our rithmetic opertors +,,,/ re deined by [u]+[v]=[u + v, u + v], [u] [v]=[u v, u v], [u] [v]= [ min{uv, uv, uv, uv}, mx{uv, uv, uv, uv} ], [u]/[v]= [ min{u/v, u/v, u/v, u/v}, mx{u/v, u/v, u/v, u/v} ], where / [v, v] We denote by R I the set o ll intervls o R,ndbyR + I nd R I the sets o ll positive intervls nd negtive intervls o R, respectively The Husdor Pompeiu distnce between intervls [u, u] nd[v, v] isdeinedby d [u, u], [v, v] = mx { u v, u v } It is well known tht R I, discompletemetricspce A division o [, b]isnyiniteorderedsubsetd hving the orm D = { = t < t < < t n = b} The mesh o division D is the mximum length o the subintervls comprising D, ie, meshd=mx{t i t i : k =,,,n} Let Dδ,[, b] be the set o ll D D[, b] such tht meshd<δinechintervl[t i, t i ], where i n, choose n rbitrry point ξ i nd orm the sum S, D, δ= ξ i t i t i, where :[, b] R or R I We cll S, D, δ Riemnn sum o corresponding to D Dδ,[, b] Deinition Aunction :[, b] R is clled Riemnn integrble R-integrble on [, b]ithereexistsa R such tht, or ech ɛ >,thereexistsδ >suchtht S, D, δ A < ɛ or every Riemnn sum S o corresponding to ech D Dδ,[, b] nd independent o thechoiceoξ i [t i, t i ]or i ninthiscse,a is clled the R-integrl o on [, b] nd is denoted by A =R t dt The collection o ll unctions tht re R-integrble on [, b]willbedenotedbyr [,b] The ollowing deinition is specil cse o the Riemnn integrl or set-vlued mps which ws erlier given by Dinghs in 956 [3]

4 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 4 o 4 Deinition Aunction :[, b] R I is clled intervl Riemnn integrble IRintegrble on [, b]ithereexistsa R I such tht, or ech ɛ >,thereexistsδ >such tht d S, D, δ, A < ɛ or every Riemnn sum S o corresponding to ech D Dδ,[, b] nd independent o the choice o ξ i [t i, t i ]or i ninthiscse,ais clled the IR-integrl o on [, b] nd is denoted by A =IR t dt The collection o ll unctions tht re IR-integrble on [, b]willbedenotedbyir [,b] Remrk 3 The concept o IR-integrl given in Deinition is equivlent to the IRintegrl given in [8, Deinition 9] The ollowing theorem ws obtined in [8] Theorem 4 An intervl-vlued unction t IR [,b] i nd only i t, t R [,b] nd IR [ t dt = R t dt,r ] t dt 3 Generlized Jensen s inequlity or intervl-vlued unctions The ollowing concepts re well known Deinition 3 We sy tht :[, b] R is convex unction i or ll x, y [, b] nd t [, ] we hve tx + ty t x+ t y I inequlity is reversed, is sid tobe concve Deinition 3 Breckner, [3] Let s, ] A unction :[, [, is clled n s-convex unction in the second sense i tx + ty t s x+ t s y or ech x, y [, ndt [, ] Deinition 33 Drgomir et l, [7] We sy tht :[, b] R is P-unction i is non-negtive nd or ll x, y [, b]ndt [, ] we hve tx + ty x+ y Deinition 34 Vrošnec, [46] Let h :[c, d] R be non-negtive unction,, [c, d]ndh We sy tht :[, b] R is n h-convex unction, or tht belongs to the clss SXh,[, b], R, i is non-negtive nd or ll x, y [, b]ndt, we hve tx + ty ht x+h t y

5 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 5 o 4 I inequlity is reversed, is sid to be h-concve, ie, SV h,[, b], R h is sid to be supermultiplictive unction i hxy hxhy 3 or ll x, y [c, d] I inequlity 3 is reversed, h is sid to be submultiplictive unction I the equlityholdsin 3, h is sid to be multiplictive unction Obviously, i ht=t, ll non-negtive convex unctions belong to SXh,[, b] nd ll non-negtive concve unctions belong to SV h,[, b] We cn introduced now the ollowing concept o unction Deinition 35 Let h :[c, d] R be non-negtive unction,, [c, d]ndh We sy tht :[, b] R + I is n h-convex intervl-vlued unction i or ll x, y [, b] nd t, we hve ht x+h t y tx + ty 4 I set inclusion 4 is reversed, is sid tobe h-concve is h-inei it is both h- concve nd h-convexthe set o ll h-convex h-concve, h-ineintervl-vlued unctionsisdenotedby SX h,[, b], R + I SV h,[, b], R + I, SA h,[, b], R + I,respectively Remrk 36 It is cler tht i ht=t s, Deinition 35 implies specil cse o convexity introduced by Breckner [4] Theorem 37 Let :[, b] R + I be n intervl-vlued unction such tht t=[t, t] Then SXh,[, b], R + I i nd only i SXh,[, b], R+ nd SV h,[, b], R + Proo Suppose tht SXh,[, b], R + I, nd consider x, y [, b], t, Then we hve ht x+h t y tx + ty, tht is, [ ht x+h t y, ht x+h t y ] [ tx + ty, tx + ty ] 5 It ollows tht ht x+h t y [ tx + ty nd ht x+h t y tx + ty

6 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 6 o 4 This shows tht SXh,[, b], R + nd SV h,[, b], R + Conversely, i SXh,[, b], R + nd SV h,[, b], R +, rom Deinition 33 nd set inclusion 5 it ollows tht SXh,[, b], R + I nd the proo is complete Theorem 38 Let :[, b] R + I be n intervl-vlued unction such tht t=[t, t] Then SV h,[, b], R + I i nd only i SV h,[, b], R+ nd SXh,[, b], R + Proo TheprooissimilrtothtoTheorem37,soweomitit Theorem 39 Vrošnec, [46] Let w, w,,w n be positive rel numbers n I h is non-negtive supermultiplictive unction nd i SXh,[, b], R +, x, x,,x n [, b], w i x i W n wi h x i, 6 W i where W n = n w i I h is submultiplictive nd SV h,[, b], R +, inequlity 6 is reversed Theorem 3 Let g R [,b] such tht g :[, b] [m, M], h : I [, be supermultiplictive unction nd :[m, M] [, be h-convex nd continuous I the ollowing limit exists, is inite, nd ht lim = k >, t + t gs ds b k b gs ds Proo Consider the division D Dδ,[, b] T begivenby D = { = t < t < < t n < t n = b}, where t i = + i n b or i n Inechintervl[t i, t i ], where i n, choose ξ i = t i = + i b nd orm the Riemnn sum n S, D, δ= ξ i t i t i Thnks to g R [,b], gs ds = lim n b gξ i t i t i = lim n n g + in b Since :[m, M] [, is continuous, the composite unction g R [,b] nd gs b ds = lim n n g + in b

7 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 7 o 4 In ddition, is h-convex, we hve b b n g + in b h n = h n b n g Consequently, we obtin Also lim n b b n ht lim = k >, t + t gs ds b k b + in b b n g + in b g + i n b = gs ds gs ds b The proo is complete Remrk 3 It is cler, i ht= t k =, nd we get the ollowing Jensen s inequlity: gs ds b b gs ds Theorem 3 Let g R [,b] such tht g :[, b] [m, M], h : I [, be submultiplictive unction nd :[m, M] [, be h-concve nd continuous I the ollowing limit exists, is inite, nd ht lim = k >, t + t gs ds b k b gs ds Proo TheprooissimilrtothtoTheorem3 nd hence is omitted As consequenceotheorems3 nd3, wehvetheollowingresult

8 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 8 o 4 Theorem 33 Let g R [,b] such tht g :[, b] [m, M], h : I [, be multiplictive unction, nd :[m, M] [, be h-ine nd continuous I the ollowing limit exists, is inite, nd ht lim = k >, t + t gs ds b = k b gs ds Theorem 34 Let g R [,b] such tht g :[, b] [m, M], h : I [, be multiplictive unction nd :[m, M] R + I be h-convex nd continuous such tht t=[t, t] I the ollowing limit exists, is inite, nd ht lim = k >, t + t k b IR gs ds R b gs ds Proo The proo is combintion o Theorems, 4, 3,nd3 Remrk 35 It is cler tht i ht=t,k =,ndwehve b IR gs ds R b gs ds I [, b] = [, ], we get the ollowing Jensen s inequlity [9, Theorem 35]: IR gs ds R gs ds It is importnt to note tht the bove Jensen s inequlity or convex set-vlued mps is due to Mtkowski nd Nikodem [4] Similrly, we cn get the ollowing theorem which gives generliztion o [9, Theorem 34] Theorem 36 Let g R [,b] such tht g :[, b] [m, M], h : I [, be multiplictive unction, nd :[m, M] R + I be h-concve nd continuous such tht t=[t, t] I the ollowing limit exists, is inite, nd ht lim = k >, t + t k b IR gs ds R b gs ds

9 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 9 o 4 NexttheoremollowsromTheorems34 nd36 Theorem 37 Let g R [,b] such tht g :[, b] [m, M], h : I [, be multiplictive unction nd :[m, M] R + I be h-ine nd continuous such tht t=[t, t] I the ollowing limit exists, is inite, nd ht lim = k >, t + t R b gs ds = k b IR gs ds 4 Hermite Hdmrd type inequlity or intervl-vlued unctions Now, the ppliction o Theorems, 4, 37,nd38 gives the ollowing result Theorem 4 Let :[, b] R + I be n intervl-vlued unction such tht t=[t, t] nd IR [,b], h : [, ] R be non-negtive unction nd h I SXh,[, b], R + I, + b h b I SV h,[, b], R + I, + b h b x dx [ +b ] ht dt x dx [ +b ] ht dt Remrk 4 It is cler tht i ht=t s, Theorem 4 reduces to the result o Osun- Gómezetl [38, Theorem 4]: + b s b x dx [ ] +b s + I ht=t, Theorem 4 reduces to the result or convex unction: + b b x dx +b I ht=, Theorem 4 reduces to the result or P-unction: + b b x dx + b I t= t, Theorem 4 reduces to the result o Sriky et l [43, Theorem 6]: + b h b x dx [ +b ] ht dt

10 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge o 4 ThenextresultgenerlizesTheorem3o[3] Theorem 43 Let :[, b] R + I be n intervl-vlued unction such tht t=[t, t] nd IR [,b], h : [, ] R be non-negtive unction nd h I SXh,[, b], R + I, where nd + b 4[h x dx ] b [ +b ][ ] + h ht dt, [ ] 3 + b +3b = 4h [ ] +b + b = + ht dt I SV h,[, b], R + I, + b 4[h x dx ] b [ +b ][ ] + h ht dt Proo The proo is completed by combining Theorems 4, 37 nd the result by Noor et l [3, Theorem 3] Exmple 44 Suppose tht [, b]=[,]letht=t or ll t [, ] nd :[, b] R + I be deined by x= [ x, e x] or ll x [, ] We hve 4[h ] = b = [ + b, 3 [ 4 x dx = [, 9 e = [, e], ] [ 5 = 4, e, 3 ], ] + [, e] e + e e ], [ ] 3 39 e e =,, 4

11 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge o 4 nd [ ] [ +b + h ] ht dt = [, ] 9 e Then we obtin tht [ 5 [, e] 4, e + e e Consequently, Theorem 43 is veriied ThenextresultgenerlizesTheorem7o[43] ] [ ] [ ] 4 e 3 39 e e,, [, 3 4 ] 9 e Theorem 45 Let, g :[, b] R + I be two intervl-vlued unctions such tht t = [ t, t], gt =[gt, gt] nd g IR [,b], h, h : [, ] R be non-negtive continuous unctions I SXh,[, b], R + I, g SXh,[, b], R + I, where b xgx dx M, b h th t dt + N, b h th t dt, M, b = g + bgb nd N, b = gb + bg Proo The proo is completed by combining Theorems 4, 37, 3 nd the result by Sriky, Sglm, nd Yildirim [43, Theorem 7] Theorem 46 Let, g :[, b] R + I be two intervl-vlued unctions such tht t = [ t, t], gt =[gt, gt] nd g IR [,b], h, h : [, ] R be non-negtive continuous unctions nd h h I SXh,[, b], R + I, g SXh,[, b], R + I, + b + b h h g b + N, b xgx dx + M, b Proo By hypothesis, one hs h h h th t dt h th t dt t + tb + h t + tb g t + tb + h g t + tb g + b + b,

12 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge o 4 Then + b g h + b h [ t + tb g t + tb + t + tb g t + tb + t + tb g t + tb + t + tb g t + tb, t + tb g t + tb + t + tb g t + tb + t + tb g t + tb + t + tb g t + tb ] = h + h + h + h = h + h h [ t + tb g t + tb, t + tb g t + tb ] [ ] h t + tb g t + tb, t + tb g t + tb [ ] h t + tb g t + tb, t + tb g t + tb [ ] h t + tb g t + tb, t + tb g t + tb [ ] h t + tb g t + tb + t + tb g t + tb [ h t + tb g t + tb + t + tb g t + tb ] [ ] h h t + tb g t + tb + t + tb g t + tb [h + h h t +h t b h tg+h tgb + h t +h t b h tg+h tgb ] [ ] h t + tb g t + tb + t + tb g t + tb = h [h + h h th t+h th t M, b + h th t+h th t N, b ] Integrting over [, ], we hve + b + b h h g b + N, b xgx dx + M, b h th t dt h th t dt This concludes the proo

13 Zho et l Journl o Inequlities nd Applictions 8 8:3 Pge 3 o 4 5 Results nd discussion We obtin some Jensen nd Hermite Hdmrd type inequlities or h-convex intervlvlued unctions Our results not only improve upon work by Cost but lso generlize the results o Sriky et l Becuse o the lck o intervl derivtives with good properties, we hve not investigted inequlities involving intervl derivtives 6 Conclusions This pper introduced the h-convex concve, ine concept or intervl-vlued unctions Under the bove concept, we presented some Jensen nd Hermite Hdmrd type inequlities or intervl-vlued unctions Our results generlize the previous inequlities presented by Cost et l The next step in the reserch direction proposed here is to investigte Jensen nd Hermite Hdmrd type inequlities or intervl-vlued unctions nd uzzy-vlued unctions on time scles Acknowledgements The uthors re very grteul to two nonymous reerees, or severl vluble nd helpul comments, suggestions nd questions, which helped them to improve the pper into present orm Funding This reserch is supported by the Fundmentl Reserch Funds or the Centrl Universities 7B974 nd 7B744 nd Nturl Science Foundtion o Jingsu Province BK85 Avilbility o dt nd mterils Not pplicble Competing interests The uthors declre tht they hve no competing interests Authors contributions All uthors contributed eqully to the writing o this pper All uthors red nd pproved the inl mnuscript Publisher s Note Springer Nture remins neutrl with regrd to jurisdictionl clims in published mps nd institutionl ilitions Received: 9 June 8 Accepted: November 8 Reerences Angulo, H, Gimenez, J, Moros, AM, Nikodem, K: On strongly h-convex unctions Ann Funct Anl, Bombrdelli, M, Vro snec, S: Properties o h-convex unctions relted to the Hermite Hdmrd Fejér inequlities Comput Mth Appl 58, Breckner, WW: Stetigkeitsussgen ür eine Klsse verllgemeinerter konvexer unktionen in topologischen lineren Räumen Publ Inst Mth 3, Breckner, WW: Continuity o generlized convex nd generlized concve set-vlued unctions Rev Anl Numér Théor Approx, Chlco-Cno, Y, Flores-Frnulič, A, Román-Flores, H: Ostrowski type inequlities or intervl-vlued unctions using generlized Hukuhr derivtive Comput Appl Mth 3, Chlco-Cno, Y, Lodwick, WA, Condori-Equice, W: Ostrowski type inequlities nd pplictions in numericl integrtion or intervl-vlued unctions Sot Comput 9, Chlco-Cno, Y, Ruián-Lizn, A, Román-Flores, H, Jiménez-Gmero, MD: Clculus or intervl-vlued unctions using generlized Hukuhr derivtive nd pplictions Fuzzy Sets Syst 9, Chlco-Cno, Y, Silv, GN, Ruián-Lizn, A: On the Newton method or solving uzzy optimiztion problems Fuzzy Sets Syst 7, Cost, TM: Jensen s inequlity type integrl or uzzy-intervl-vlued unctions Fuzzy Sets Syst 37, Cost, TM, Bouwmeester, H, Lodwick, WA, Lvor, C: Clculting the possible conormtions rising rom uncertinty in the moleculr distnce geometry problem using constrint intervl nlysis In Sci 45 46, Cost, TM, Chlco-Cno, Y, Lodwick, WA, Silv, GN: Generlized intervl vector spces nd intervl optimiztion In Sci 3, Cost, TM, Román-Flores, H: Some integrl inequlities or uzzy-intervl-vlued unctions In Sci 4, Dinghs, A: Zum Minkowskischen Integrlbegri bgeschlossener Mengen Mth Z 66, Drgomir, SS: Inequlities o Hermite Hdmrd type or h-convex unctions on liner spces Proyecciones 34,

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