Integral inequalities via fractional quantum calculus

Transcription

1 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 DOI.86/s R E S E A R C H Open Access Integrl ineulities vi frctionl untum clculus Weerwt Sudsutd, Sotiris K Ntouys,3 nd Jessd Triboon * * Correspondence: jessd.t@sci.kmutnb.c.th Nonliner Dynmic Anlysis Reserch Center, Deprtment of Mthemtics, Fculty of Applied Science, King Mongkut s University of Technology North Bngkok, Bngkok, 8, Thilnd Full list of uthor informtion is vilble t the end of the rticle Abstrct In this pper we prove severl frctionl untum integrl ineulities for the new -shifting opertor mm + introduced in Triboon et l. Adv. Differ. Eu. 5:8, 5, such s: the-hölder ineulity, the -Hermite-Hdmrd ineulity, the -Cuchy-Bunykovsky-Schwrz integrl ineulity, the -Grüss integrl ineulity, the -Grüss-Čebyšev integrl ineulity, nd the -Póly-Szegö integrl ineulity. MSC: 5A3; 6D; 6A33 Keywords: frctionl integrl; frctionl integrl ineulities; untum clculus Introduction The untum clculus is known s the clculus without limits. It substitutes the clssicl derivtive by difference opertor, which llows one to del with sets of nondifferentible functions. Quntum difference opertors hve n interesting role due to their pplictions in severl mthemticl res, such s orthogonl polynomils, bsic hypergeometric functions, combintorics, the clculus of vritions, mechnics, nd the theory of reltivity. The book by Kc nd Cheung []covers mny of the fundmentlspectsof untum clculus. In recent yers, the topic of -clculus hs ttrcted the ttention of severl reserchers nd vriety of new results cn be found in the ppers [3 5] nd the references cited therein. In [6] the notions of k -derivtive nd k -integrl of continuous function f : [t k, t k+ ] R, hve been introduced nd their bsic properties were proved. As pplictions existence nd uniueness results for initil vlue problems of first nd second order impulsive k -difference eutions were investigted. The -clculus nlogs of some clssicl integrl ineulities, such s Hölder, Hermite-Hdmrd, Trpezoid, Ostrowski, Cuchy-Bunykovsky-Schwrz, Grüss nd Grüss-Čebyšev were estblished in [7]. For recent results on untum ineulities, see [8 ]. In [] new concepts of frctionl untum clculus were defined, by defining new -shifting opertor m m +. After giving the bsic properties the - derivtive nd -integrl were defined. New definitions of the Riemnn-Liouville frctionl -integrl nd the -difference on n intervl [, b] were given nd their bsic properties were discussed. As pplictions of the new concepts, one proved existence nd 6 Sudsutd et l. This rticle is distributed under the terms of 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 if chnges were mde.

2 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge of 5 uniueness results for first nd second order initil vlue problems for impulsive frctionl -difference eutions. In this pper we prove severl integrl ineulities for the new -shifting opertor m m +, suchs:the-hölder ineulity, the -Hermite-Hdmrd ineulity, the -Korkine integrl eulity, the -Cuchy-Bunykovsky-Schwrz integrl ineulity, the -Grüss integrl ineulity, the -Grüss-Čebyšev integrl ineulity, nd the -Poly-Szegö integrl ineulity. Preliminries To mke this pper self-contined, below we recll some well-known fcts on frctionl -clculus. The presenttion here cn be found, for exmple, in [7, 8]. Let us define -shifting opertor s mm +,. where < <,m, R. For ny positive integer k,wehve k m k m nd mm.. The following results cn be found in []. Property. For ny m, n R nd for ll positive integer k, j, the following properties hold: i k m k m; ii j k m k j m j+k m; iii ; iv k m k m ; v m k m k m ; vi k mm m k,form ; vii m k nm k n. The -nlog of the Pochhmmer symbol is defined by k m;, m; k i m, k N { }..3 We lso define the power of the -shifting opertor s n m, n m k More generlly, if γ R,then n m γ n γ k n i m, k N { }..4 n i m/n n γ +i, n..5 m/n From the bove definitions, the following results were proved in [].

3 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 3 of 5 Property. For ny γ, m, n R with n nd k N { }, the following properties hold: i n m k ii n m γ n k m n ; k; n γ m n i m iii n k nγ n γ k ; γ +k ;. The -number is defined by n γ +i n γ m n ; m n γ ; ; [m] m, m R..6 If ndm n,then.5 is reduced to γ The -gmm function is defined by i+..7 γ +i+ Ɣ t t, t R\{,,,...}..8 t Obviously, Ɣ t +[t] Ɣ t. For ny s, t >,the-bet function is defined by B s, t u s u t d u..9 The -bet function in terms of the -gmm function cn be written s B s, t Ɣ sɣ t Ɣ s + t.. Let usgive the definitionsof Riemnn-Liouville frctionl -integrl nd the -derivtive on the dense intervl [, b]. Definition.3 Let α ndf be continuous function defined on [, b]. The frctionl -integrl of Riemnn-Liouville type is given by I f tftnd I α f t t t s α f s Ɣ α d s t i t i+ Ɣ α t α f i t. Definition.4 The frctionl -derivtive of Riemnn-Liouville type of order α of continuous function f on the intervl [, b]isdefinedby D f tftnd D α f t D υ I υ α f t, α >, where υ is the smllest integer greter thn or eul to α.

4 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 4 of 5 Lemm.5 [] Let α, β, nd f be continuous function on [, b]. The Riemnn- Liouville frctionl -integrl hs the following semi-group properties: I β I α f t I α I β f t I α+β f t.. Throughoutthis pper, in someplces, thevrible s will be shown inside the frctionl integrl nottion s I α f st, which mens I α f s t t t s α f s Ɣ α d s. Lemm.6 If α, β, then, for t [, b], the following reltion holds: I α s β t Ɣ β + Ɣ β + α + t β+α.. Proof From Definition.3 nd pplying Property.iv, Property.iii, it follows tht I α s β t t Ɣ α t Ɣ α t Ɣ α t s α s β d s t β+α Ɣ α t β+α Ɣ α t β+α Ɣ α t β+α Ɣ α t β+α Ɣ α i t i+ t α i t β i t α i+ ; α+i ; i t β i i+ ; α+i ; βi i i+ i i+ α +i βi i i t β+α B β +,α Ɣ α Ɣ β + Ɣ β + α + t β+α, i+ i+ s β s α d s α βi α i β which leds to.sreuired. Corollry.7 Let f ttndgtt for t [, b], nd α >.Then we hve i I α t α f st Ɣ α+ t +[α +] ; ii I α t α gst Ɣ α+3 + t +t [α +] + [α +] [α +].

5 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 5 of 5 3 Min results Let us strt with the frctionl -Hölder ineulity on the intervl [, b]. Theorem 3. Let < <,α >,p, p >,such tht p + p.then for t [, b] we hve I α f s gs t I α f s p t p I α gs p t p. 3. Proof From Definition.3 nd the discrete Hölder ineulity, we hve I α f s gs t t t s α Ɣ α f s gs d s t i t i+ Ɣ α t α f i t g i t t Ɣ α t Ɣ α t Ɣ α t i+ i t i+ i t α i p f i t i p g i t t i+ t α t α t t s α f s p p Ɣ α d s t t s α gs p Ɣ α d s I α f s p t p I α gs p t p. f i t p p g i t p p p Therefore, ineulity 3.holds. Remrk 3. If α nd,then3. is reduced to the -Hölder ineulity in []. The frctionl -Hermite-Hdmrd integrl ineulity on the intervl [, b] will be proved s follows. Theorem 3.3 Let f :[, b] R be convex continuous function,< <nd α >.Then we hve + b Ɣ α + f I α b α f + b s b I α b α f s b [α +] f +fb. 3. Ɣ α +

6 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 6 of 5 Proof The convexity of f on [, b]menstht f s + sb sf +sf b, s [, ]. 3.3 Multiplying both sides of 3.3by s α /Ɣ α, s,, we get Ɣ α s α f s + sb f Ɣ α s α s+ f b Ɣ α s α s. 3.4 Tking -integrtion of order α >for3.4withrespecttos on [, ], we hve Ɣ α f Ɣ α which mens tht s α f s + sb d s s α s d s + f b s α s Ɣ α d s, 3.5 I α f s + sb f I α s + f b I α s. 3.6 From Corollry.7i, we hve I α s Ɣ α + nd I α s Ɣ α + Ɣ α +. Using the definition of frctionl -integrtion on [, b], we hve I α f s + sb Ɣ α Ɣ α Ɣ α s α f s + sb d s i i i+ α f i + i b b α Ɣ α i i+ i α+i f i b i b α i+ ; i+α ; f i b b b α b s α f s Ɣ α d s I α b α f b, which gives the second prt of 3.byusing3.6.

7 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 7 of 5 To prove the first prt of 3., we use the convex property of f s follows: [ ] s + sb + s + sb f s + sb + f s + sb f + b f. 3.7 Multiplying both sides of 3.7by s α /Ɣ α, s,, we get + b f Ɣ α s α Ɣ α s α f s + sb + Ɣ α s α f s + sb. Agin on frctionl -integrtion of order α > to the bove ineulity with respect to t on [, ] nd chnging vribles, we get + b f Ɣ α + I α b α f s b+ Ɣ α + I α b α f + b s b. 3.8 By direct computtion, we hve I α f sb + s Ɣ α Ɣ α Ɣ α i i+ α f i b + i i b α Ɣ α i+ α f + b i b i i+ ; α+i ; f + b i b I α b α f + b s b, b s α f + b s d s together with 3.8,we derivethefirstprtofineulity3. s reuested. The proof is completed. Remrk 3.4 If α nd, then ineulity 3. is reduced to the clssicl Hermite- Hdmrd integrl ineulity s + b f b See lso [, 3]. f s ds f +fb.

8 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 8 of 5 Let us prove the frctionl -Korkine eulity on the intervl [, b]. Lemm 3.5 Let f, g :[, b] R be continuous functions, < <,nd α >.Then we hve I α f s fr gs gr b b α I α Ɣ α + f sgs b I α f s b I α gs b. 3.9 Proof From Definition.3,we hve I α f s fr gs gr b Ɣ α b s α b r α f s f r gs gr d s d r Ɣ α b s α b r α f sgs fsgr frgs+frgr d s d r b α b i b i+ Ɣ α + Ɣ α b α f i b g i b b i b i+ Ɣ α b i b i+ Ɣ α b i b i+ Ɣ α b i b i+ Ɣ α b α b + i Ɣ α + Ɣ α b α b i Ɣ α + Ɣ α b i b i+ Ɣ α b Ɣ α i b i+ b α f i b b α g i b b α g i b b α b i+ b i+ f i b b α f i b g i b b α b α f i b b α f i b g i b g i b b α b b s α f sgs Ɣ α + Ɣ α d s

9 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 9 of 5 Ɣ α Ɣ α b s α f s d s b s α gs d s b α I α Ɣ α + fg b I α f b I α g b, from which one deduces 3.9. Remrk 3.6 If α, then Lemm 3.5 is reduced to Lemm 3. in [7]. Next,wewillprovethefrctionl-Cuchy-Bunykovsky-Schwrz integrl ineulity on the intervl [, b]. Theorem 3.7 Let f, g :[, b] R be continuous functions, < <,nd α, β >.Then we hve I β+α f s, rgs, r b I β+α f s, r b I β+α g s, r b. 3. Proof From Definition.3,wehve I β+α f s, r b Ɣ αɣ β b Ɣ αɣ β f i b, n b. b s α n i+n b i+ b r β f s, r d s d r b α b i+ Using the clssicl discrete Cuchy-Schwrz ineulity, we hve I β+α f s, rgs, r b b Ɣ αɣ β n i+n b i+ f i b, n b g i b, n b b Ɣ αɣ β n i+n b i+ f i b, n b b Ɣ αɣ β b i+ b β b α b α g i b, n b I β+α f s, r b I β+α g s, r b. n b i+ b i+ i+n b β b β b β b i+ b α Therefore, ineulity 3.holds.

10 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge of 5 Remrk 3.8 If α,thenineulity3. is reduced to the -Cuchy-Bunykovsky- Schwrz integrl ineulity in [7]. Now, we will prove the frctionl -Grüss integrl ineulity on the intervl [, b]. Theorem 3.9 Let f, g :[, b] R be continuous functions stisfying φ f s, ψ gs, for ll s [, b], φ,, ψ, R. 3. For < <nd α >,we hve the ineulity Ɣ α + I α b α f sgs Ɣ α + b b α I α f s b Ɣ α + I α b α gs b φ ψ Proof Applying Theorem 3.7,wehve I α f s fr gs gr b I α From Lemm 3.5, it follows tht f s fr b I α gs gr b. 3.3 I α b α f s fr b I α Ɣ α + f s b I α f s b. 3.4 By simple computtion, we hve Ɣ α + I α b α f s Ɣ α + b Ɣ α + b α I α f s b I α b α f s b Ɣ α + b α I α f s b φ Ɣ α + b α I α f s φ f s b, 3.5 nd n nlogous identity for g. By ssumption 3.wehvefs φ f s for ll s [, b], which implies I α f s φ f s b. From 3.5ndusingthefcttht A+B AB, A, B R,wehve Ɣ α + I α b α f s Ɣ α + b Ɣ α + b α I α f s b I α b α f s b Ɣ α + b α I α f s b φ

11 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge of 5 4 [ Ɣ ] α + I α b α f s Ɣ α + b + I α b α f s b φ 4 φ. 3.6 A similr rgument gives I α gs gr b 4 ψ. 3.7 Using ineulity 3.3vi3.4 nd the estimtions 3.6nd3.7, we get I α b α f s fr gs gr b φ ψ. 4 Therefore, ineulity 3.holds,sdesired. Remrk 3. If α nd, then ineulity 3. is reduced to the clssicl Grüss integrl ineulity s b f sgs ds b φ ψ. 4 f s ds b gs ds See lso [, 3]. Next, we re going to prove the frctionl -Grüss-Čebyšev integrl ineulity on the intervl [, b]. Theorem 3. Let f, g :[, b] R be L -, L -Lipschitzin continuous functions, so tht f s f r L s r, gs gr L s r, 3.8 for ll s, r [, b], < <,L, L >,nd α >.Then we hve the ineulity b α I α Ɣ α + f sgs b I α f s b I α gs b L L b α+ + [α +] [α +]. 3.9 Ɣ α +Ɣ α +3 Proof Recll the frctionl -Korkine eulity s b α I α Ɣ α + f sgs b I α f s b I α gs b I α f s fr gs gr b. 3. It follows from 3.8tht f s f r gs gr L L s r, 3.

12 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge of 5 for ll s, r [, b]. Tking the double frctionl -integrtion of order α with respect to s, r [, b], we get I α f s fr gs gr b Ɣ α b s α b r α f s f r gs gr d s d r L L Ɣ α L L Ɣ α L L Ɣ α + L L Ɣ α b s α b s α b s α b r α s r d s d r b r α s d s d r b s α b α L L I α Ɣ α + s b I α s b b r α sr d s d r b r α r d s d r. 3. From Corollry.7ii, with t b,wehve I α s b b α + b +b [α +] + [α +] [α +]. Ɣ α +3 By direct computtion, we hve b α I α Ɣ α + s b I α s b b α + b +b [α +] + [α +] [α +] Ɣ α +Ɣ α +3 b α Ɣ α + b + [α +] b α+ + [α +] [α +]. 3.3 Ɣ α +Ɣ α +3 Thus, from 3.nd3.3, we hve I α f s fr gs gr b L L b α+ + [α +] [α +]. 3.4 Ɣ α +Ɣ α +3 By pplying 3.4to3., we get the desired ineulity in 3.9.

13 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 3 of 5 Remrk 3. If α nd, then ineulity 3.9 is reduced to the clssicl Grüss- Čebyšev integrl ineulity s b See lso [, 3]. f sgs ds b f s ds b gs ds L L b. For the finl result, we estblish the frctionl -Póly-Szegö integrl ineulity on the intervl [, b]. Theorem 3.3 Let f, g :[, b] R be two positive integrble functions stisfying <φ f s, <ψ gs, for ll s [, b], φ,, ψ, R Then for < <nd α >,we hve the ineulity I αf sb I αg sb I αf φψ sgsb φψ Proof From 3.5, for s [, b], we hve φ f s gs ψ, which yields ψ f s 3.7 gs nd f s gs φ. 3.8 Multiplying 3.7nd3.8, we obtin or ψ f s f s gs gs φ, ψ + φ f s gs f s g s + φ ψ. 3.9 Ineulity 3.9cnbewrittens φψ + f sgs ψ f s+φ g s. 3.3

14 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 4 of 5 Multiplying both sides of 3.3by b s α /Ɣ α nd integrting with respect to s from to b,weget φψ + I α f sgs b ψ I α f s b+φ I α g s b. Applying the AM-GM ineulity, A + B AB, A, B R +,wehve φψ + I α f sgs b φψ I α f s b I α g s b, which leds to φψ I α f s b I α g s b 4 φψ + I α f sgs b. Therefore, ineulity 3.6isproved. Remrk 3.4 If α nd, then ineulity 3.6 is reduced to the clssicl Póly- Szegö integrl ineulity s f s ds g s ds f sgs φψ ds 4 +. φψ See lso [4]. 4 Conclusion In this work, some importnt integrl ineulities involving the new -shifting opertor mm +, introducedin[], re estblished in the context of frctionl untum clculus. The derived results constitute contributions to the theory of integrl ineulities nd frctionl clculus nd cn be specilized to yield numerous interesting frctionl integrl ineulities including some known results. Furthermore, they re expected to led to some pplictions in frctionl boundry vlue problems. Competing interests The uthors declre tht they hve no competing interests. Authors contributions All uthors contributed eully to this rticle. They red nd pproved the finl mnuscript. Author detils Nonliner Dynmic Anlysis Reserch Center, Deprtment of Mthemtics, Fculty of Applied Science, King Mongkut s University of Technology North Bngkok, Bngkok, 8, Thilnd. Deprtment of Mthemtics, University of Ionnin, Ionnin, 45, Greece. 3 Nonliner Anlysis nd Applied Mthemtics NAAM-Reserch Group, Deprtment of Mthemtics, Fculty of Science, King Abdulziz University, P.O. Box 83, Jeddh, 589, Sudi Arbi. Received: 8 November 5 Accepted: 8 Februry 6 References. Triboon, J, Ntouys, SK, Agrwl, P: New concepts of frctionl untum clculus nd pplictions to impulsive frctionl -difference eutions. Adv. Differ. Eu. 5, 8 5. Kc, V, Cheung, P: Quntum Clculus. Springer, New York 3. Jckson, FH: -Difference eutions. Am. J. Mth. 3, Al-Slm, WA: Some frctionl -integrls nd -derivtives. Proc. Edinb. Mth. Soc. 5, / Agrwl, RP: Certin frctionl -integrls nd -derivtives. Proc. Cmb. Philos. Soc. 66, Ernst, T: The history of -clculus nd new method. UUDM Report :6, Deprtment of Mthemtics, Uppsl University

15 Sudsutd et l. Journl of Ineulities nd Applictions 6 6:8 Pge 5 of 5 7. Ferreir, R: Nontrivil solutions for frctionl -difference boundry vlue problems. Electron. J. Qul. Theory Differ. Eu., 7 8. Annby, MH, Mnsour, ZS: -Frctionl Clculus nd Eutions. Lecture Notes in Mthemtics, vol. 56. Springer, Berlin 9. Bngerezko, G: Vritionl -clculus. J. Mth. Anl. Appl. 89, Ismil, MEH, Simeonov, P: -Difference opertors for orthogonl polynomils. J. Comput. Appl. Mth. 33, Yu, C, Wng, J: Existence of solutions for nonliner second-order -difference eutions with first-order -derivtives. Adv. Differ. Eu. 3, 4 3. Ahmd, B, Ntouys, SK: Boundry vlue problems for -difference inclusions. Abstr. Appl. Anl., Ahmd, B: Boundry-vlue problems for nonliner third-order -difference eutions. Electron. J. Differ. Eu., Gref, JR, Kong, L: Positive solutions for clss of higher-order boundry vlue problems with frctionl -derivtives. Appl. Mth. Comput. 8, Ahmd, B, Ntouys, SK, Purnrs, IK: Existence results for nonlocl boundry vlue problems of nonliner frctionl -difference eutions. Adv. Differ. Eu., 4 6. Triboon, J, Ntouys, SK: Quntum clculus on finite intervls nd pplictions to impulsive difference eutions. Adv. Differ. Eu. 3, Triboon, J, Ntouys, SK: Quntum integrl ineulities on finite intervls. J. Ineul. Appl. 4, 4 8. Noor, MA, Noor, KI, Awn, MU: Some untum estimtes for Hermite-Hdmrd ineulities. Appl. Mth. Comput. 5, Noor, MA, Noor, KI, Awn, MU: Some untum integrl ineulities vi preinvex functions. Appl. Mth. Comput. 69, Tf, S, Brhim, K, Rihi, L: Some results for Hdmrd-type ineulities in untum clculus. Mtemtiche LXIX, Anstssiou, GA: Intelligent Mthemtics: Computtionl Anlysis. Springer, New York. Cerone, P, Drgomir, SS: Mthemticl Ineulities. CRC Press, New York 3. Pchptte, BG: Anlytic Ineulities. Atlntis Press, Pris 4. Póly, G, Szegö, G: Aufgben und Lehrstze us der Anlysis, Bnd. Die Grundlehren der mthemtischen Wissenschften, vol. 9. Springer, Berlin 95