Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings

Muhammad Bilal Khan,Hatim Ghazi Zaini,Mohamed S Soliman,Kamsing Nonlaopon,Savin Treanțǎ

doi:10.3390/math10040611

Abstract

The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions (IV-Fs) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs). Then, we obtain Hermite–Hadamard (𝓗-𝓗) and Hermite–Hadamard–Fejér (𝓗-𝓗-Fejér) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results.

Highlights

The collection of all Riemann integrable real valued functions and Riemann integrable IV-Fs is denoted by R[μ,υ] and I R[μ,υ], respectively
The following outcome is very important in the field of interval-valued calculus because, by using this result, we can handle IV-Fs
Theorem 2 establishes the relation between IV-F Y(ω) and lower function Y∗(ω) and upper function Y∗(ω)

Summary

Introduction

SoSmome eNNewewVVeSPreroSPserimrosoienimneoivnsneNevosxNeefowxFeIfuwnFVInutneVecntgtreecisrgotraiisronloainsoInlnsnisInenosiqnfetouhqIftnaeuhIltnaeIientltgIietentregisaterrelvfsaroIalvfnrloIa-enLrVlq-eeLVauqfeltaauuflaltaeiuntladeidnetdSidsReeSfsRitoegtfiitrhongtiLrthgneLstgfestfat CitaCtiiotnat:iKonh:aKn,hMan.B, M.; T.Br.e;aTnrțeǎa,nSț.;ǎ, S.; PrPerienivnevxexFuFnucntciotinosnisninthteheInItnetrevravla-Vl-ValauleudedSeStetitntignsgs SolimSoalnim, Man.S, .M; N.So.;nNlaoonploano,pKon.;, K.; eraeberlearabablselseoaascsiaostocicioainatitotihonanththhaaatsthehamasseermgmeerdgrgbeededetwbrbaeeebtewtrlnweaebatelhsenesenoatthtscwhseiaoeotcwitiownaontoriiotenihncnaerternhechtaceayetnsenhtaetayrmysese,aeawrmsrge,se,wrdmwgebeaedmyemtwbaapyeeytpewalanpyepptewphlnylehytwathwtehwhoatwateitnwolwereiaenerlcenlereaenarctrnenynetayres,awrse, wm fromfrforomnmeootnoneetthoteotohthteheoeotrh.thIener.rt.IhnIinsthtphiasipspeparap, poeeru,rrf,oroauoiurmfraoaoimsinmteoisitnesotesottothaeobestslohiatstabehbloeiltsitrhsh.ehIetnrhte.hetlIheanritersitelohaplnaitasitbopiponeantrwbp, beoeetruewt,rnwoeauieinernmtnaeiingimsnrteatoegilsgrienatrsoaletlaqienibusnetlaieaqslqbuihtulaiiteaslhishlteiitteirheseeslarteiloantibonetwb andaanindtdeinrinvteaterl-vrvaal-lluv-vaealdulufeueddnfcuftuinoncnctistoio(nInVss-(FI(VIsV)-F-bFsa)sns)bedbadnaisnuesdtedpeidonruvnutpeapotrlh-vonvenaatlph-tluvhseeapuldpusdesfoueud-undofdcourtdo-ino-ecornrtdrisrdoee(nerIlarVsrter-(iFeIloaVlsnat)-i.tFobiFosnai)nr.sbse.FtadFilrsyisuer,tsdpwltyoluye,np,wdwotihenseecdtdupihsissescsecuputshsdsesouth-tdoheored-oerrdreerlarteilo propppreroCroptiptieaeeCtrsritiotioinaeeft:sisKoloenohf:aftKnlea,hfMnatnd.aB, Mn.r;idT.gBrh.er;aitTgnrpțheǎrat,tenSppiț.;ǎnrr,eveSii.en;nxvveienxxtpienirntovteppearrvleov-ravpatlai-leev-lrsuvataieloeuldfsuelfedoeudffntulfeacuntfnintcodtacintnorisinodg(snhlres(itlfge(tphlfearttfenatpidnnardvenriedrngxihvgrihetngxtpthperitnrerpveitneiarnrvelv-ievnaexavxl-lIeuvVIxVae--FldIuFVssfe)-u).Fd.TnsTf)hcu.hteTinenonhcn,t,eiswno(n,elesf(tleafntdanridg wewoobebSttooaalbimnStoaHlnimn,eMarHnm.S,e.Mi;rtNm.eSo–.i;nHtNleaoa–ondpHloaaanomd,pKoan.mr;,dKa.(r;Hd -(H)-)Hwaanne)ddwaoHnbedteoarHbimnteaiHritmnee–riHHtmeea–irtdHmea–aimHtdeaa–mrdHdaaa–mrdFdea–jmréFdrea(jréHdr -(H)-FHaen)-jédFareH))njédtteyryrH)ppmteeyiritipmnenee–eiqHtqienuua–eaadHqlliauitatmyaydlaitmnrydda–rFde–jéFrej(éHr. AbBioraax;bs1ioa1l;0ism9o9al,inTm@aaitfnu2@.e1td9u4u.e4.sd,aSua.suadi ArabSiaa;usSdoailuiAmdraianAb@iraat;ubs.ioeadl;iusmo.salainm@atnu@.etduu.e.sdau.sa 4 D4 4epDaerDtpmeapretanmrttemonfetMnotfaotMhf eMamthaatehtmiecmas,tiaFctasi,csuF,alFtcyaucolutfylSt4oycfioeSDnf4ccSeiepcD,niaeKcernetphcm,aoeKren,thnKmKothenaoonefnKntMaoKUefaantMehinUveamenUthriasnveitimetivycreas,ri,tKtsiFycihats,yo,cK,nuFKhalKtohcynauoeolnKtnfyKaS4eoca0nfie0eSn04nc20c4i,0e0T0,n02Khc0,e2ahT,iolKhTananhiKdaolainalnaeKdnadUeniUvenrisvietyrs, iKtyh,oKnhKona MathMeInmattahetgeimcrsaal2tiI0cn2se22q,0u92,a2xl,i.t9ie, sx.for Left and httphsR:t/t/ipdgsho:/it/.odPrrogei/.ion1r0vg.e3/x3109F0.u3/3nx9xc0txi/xoxxnxsxxinx the Interval-Valued Settings. B.Oo.xB1o1x09191,0T99a,ifT2a1i9f 4241,944, SauSdaiuAdriaAbiraa;bsioal;imsoalinm@atnu@.etdu4u.e.sdDa4ue.psDaaertpmaretnmt eonf tMoaf tMheamthaetmicsa,tiFcasc, uFlatcyuoltfySocifeSnccie,nKceh,oKnhKoaneKnaUeniUvenrisvietyrs, iKtyh,oKnhKoaneKna4e

Results

Conclusion