In this paper, we improve and further generalize some Ostrowski-Grüss type inequalities for the fractional integrals by using new Montogomery identities

Basak, Nagihan

Sarikaya, Mehmet Zeki

Yaldiz, Hatice

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXIX (2014) – Fasc. I, pp. 227–235doi: 10.4418/2014.69.1.17NEW FRACTIONAL INEQUALITIESOF OSTROWSKI-GRU¨SS TYPEM. ZEKI SARIKAYA - H. YALDIZ - N. BASAKIn this paper, we improve and further generalize some Ostrowski-Gru¨ss type inequalities for the fractional integrals by using new Mont-gomery identities.1. Introduction and preliminary resultsThe inequality of Ostrowski provides us an estimates for the deviation of thevalues of a smooth function from its mean value. More precisely, if f : [a,b]→Ris a differentiable function with bounded derivative, then∣∣∣∣∣∣ f (x)− 1b−ab∫af (t)dt∣∣∣∣∣∣≤[14+(x− a+b2 )2(b−a)2](b−a)∥∥ f ′∥∥∞for every x ∈ [a,b]. Moreover the constant 1/4 is the best possible. For somegeneralizations of this classical inequality see the book [12, p.468-484] by Mitri-novic, Pecaric and Fink. A simple proof of this inequality can be done by usingthe following identity [12]: If f : [a,b]→ R is differentiable on (a,b) with thefirst derivative f ′ integrable on [a,b], then Montgomery identity holds:f (x) =1b−ab∫af (t)dt+b∫aP1(x, t) f ′(t)dt,Entrato in redazione: 11 aprile 2013AMS 2010 Subject Classification: 26D15, 41A55, 26D10, 26A33.Keywords: Montgomery identity, fractional integral, Ostrowski inequality, Gru¨ss inequality.228 MEHMET ZEKI SARIKAYA - HATICE YALDIZ - NAGIHAN BASAKwhere P1(x, t) is the Peano kernel defined byP1(x, t) :=t−ab−a , a≤ t < xt−bb−a , x≤ t ≤ b.This inequality gives an upper bound for the approximation of integral mean ofa function f by the functional value f (x) at x∈ [a,b]. In 2001, Cheng [1] provedthe following Ostrowski-Gru¨ss type integral inequality.Theorem 1.1. Let I ⊂ R be an open interval, a,b ∈ I, a < b. If f : I→ R is adifferentiable function such that there exist constants γ,Γ ∈R, with γ ≤ f ′ (x)≤Γ, x ∈ [a,b]. Then have∣∣∣∣∣∣12 f (x)− (x−b) f (b)− (x−a) f (a)2(b−a) − 1b−ab∫af (t)dt∣∣∣∣∣∣ (1)≤ (x−a)2+(b− x)28(b−a) (Γ− γ) , for all x ∈ [a,b] .Theorem 1.1 is a generalization of the following Ostrowski-Gru¨ss type in-tegral inequality, which was firstly by Dragomir and Wang in [2] and furtherimproved by Matic et al. in [6].Theorem 1.2. Let the assumptions of Theorem 1.1 hold. Then for all x ∈ [a,b],we have∣∣∣∣∣∣ f (x)− f (b)− f (a)b−a(x− a+b2)− 1b−ab∫af (t)dt∣∣∣∣∣∣≤ 14 (b−a)(Γ− γ) (2)The above two inequalities are both connections between the Ostrowski inequal-ity [4] and the Gru¨ss inequality [5] and can be applied to bound some specialmean and some numerical quadrature rules. During the past few years manyresearchers have attracted considerable attention to the above inequalities andvarious generalizations, extensions and variants of these inequalities have ap-peared in the literature, see [1], [2], [6], [24] and the references cited therein.For recent results and generalizations regarding Ostrowski and Gru¨ss inequali-ties, we refer the reader to the recent papers [1]-[6], [12]-[15], [17]-[24]. Now,we will give some necessary definitions and mathematical preliminaries of frac-tional calculus theory which are used further in this paper. More details, onecan consult [11], [16].NEW FRACTIONAL INEQUALITIES 229Definition 1.3. The Riemann-Liouville fractional integral operator of order α ≥0 with a≥ 0 is defined asJαa f (x) =1Γ(α)x∫a(x− t)α−1 f (t)dt,J0a f (x) = f (x).The theory of fractional calculus has known an intensive development overthe last few decades. It is shown that derivatives and integrals of fractional typeprovide an adequate mathematical modelling of real objects and processes see([7], [8], [9], [10], [20], [21], [22]). Therefore, the study of fractional differ-ential equations need more developmental of inequalities of fractional type. In[7] and [20], the authors established some inequalities for differentiable map-pings which are connected with Ostrowski type inequality by used the Riemann-Liouville fractional integrals, and they used the following lemma to prove theirresults:Lemma 1.4. Let f : I ⊂ R→ R be differentiable function on I◦ with a,b ∈ I =[a,b] (a < b) and f ′ ∈ L1[a,b], then, for α ≥ 1f (x) =Γ(α)b−a (b−x)1−αJαa f (b)−Jα−1a (P2(x,b) f (b))+Jαa (P2(x,b) f′(b)) (3)where P2(x, t) is the fractional Peano kernel defined byP2(x, t) =t−ab−a(b− x)1−αΓ(α), a≤ t < xt−bb−a(b− x)1−αΓ(α), x≤ t ≤ b.In this article, we use the Riemann-Liouville fractional integrals to estab-lish some new integral inequalities of Ostrowski-Gru¨ss type. From our results,the classical Ostrowski-Gru¨ss type inequalities can be deduced as some specialcases.2. Main ResultsLemma 2.1. Let f : [a,b]⊂R→R be a differentiable function on (a,b) (a < b),α ≥ 1 and f ′ ∈ L1[a,b], then the generalization of the Montgomery identity for230 MEHMET ZEKI SARIKAYA - HATICE YALDIZ - NAGIHAN BASAKfractional integrals holds:f (x) = (α+1)Γ(α)(b− x)1−α(b−a) Jαa f (b)−Jα−1a (P2(x,b) f (b))− (b− x)2−α(b−a) Γ(α)Jα−1a f (b)−(b− x)1−α (x−a)(b−a)2−α f (a)+2Jαa(K1 (x,b) f′(b)), (4)where K1 (x, t) is the fractional Peano kernel defined byK1 (x, t) :=(t− a+ x2)(b− x)1−αb−a Γ(α) , a≤ t < x(t− b+ x2)(b− x)1−αb−a Γ(α) , x≤ t ≤ b.(5)Proof. By definition of K1 (x, t), we haveJαa(K1 (x,b) f′(b))=1Γ(α)b∫a(b− t)α−1 K1 (x, t) f ′ (t)dt=(b− x)1−αb−a x∫a(b− t)α−1(t− a+ x2)f′(t)dt+b∫x(b− t)α−1(t− b+ x2)f′(t)dtthat can be written asJαa(K1 (x,b) f′(b))=12Jαa (P2(x,b) f′(b))+(b− x)1−α2(b−a)b∫a(b− t)α−1 (t− x) f ′ (t)dt. (6)For term in the right hand side of (6) integrating by parts implies thatb∫a(b− t)α−1 (t− x) f ′ (t)dt= (b− x)b∫a(b− t)α−1 f ′ (t)dt−b∫a(b− t)α f ′ (t)dt (7)= (x−a)(b−a)α−1 f (a)+(b− x)Γ(α)Jα−1a f (b)−Γ(α+1)Jαa f (b)NEW FRACTIONAL INEQUALITIES 231Substituting Jαa (P2(x,b) f′(b)) in the Lemma 1.4 and (7) in (6), we obtain thatJαa(K1 (x,b) f′(b))=12f (x)− (α+1)Γ(α)(b− x)1−α2(b−a) Jαa f (b)+12Jα−1a (P2(x,b) f (b))+(b− x)1−α (x−a)2(b−a)α−2 f (a)+ (b− x)2−α2(b−a) Γ(α)Jα−1a f (b) .The proof is completed.Remark 2.2. Letting α = 1, formula (4) reduces the following identity12f (x) =1(b−a)b∫af (t)dt+(x−b) f (b)− (x−a) f (a)2(b−a) +b∫aK (x, t) f′(t)dtwhereK (x, t) :=(t− a+ x2), t ∈ [a,x)(t− b+ x2), t ∈ [x,b]which was given by Tong and Guan in [24]. Using the above identity, the authorsproved another simple proof of Theorem 1.1.Now using the new Montgomery identity for fractional integrals (4) and thecorresponding fractional Peano kernel (5), we derive a new Ostrowski-Gru¨sstype inequality of fractional type.Theorem 2.3. Let f : [a,b]⊂ R→ R be a differentiable function on (a,b) (a <b) and | f ′ (x)| ≤M for any x ∈ [a,b]. Then the following fractional inequalityholds: ∣∣∣∣∣12 f (x)− (α+1)Γ(α)(b− x)1−α2(b−a) Jαa f (b)+ 12Jα−1a (P2(x,b) f (b))+(b− x)2−α2(b−a) Γ(α)Jα−1a f (b)+(b− x)1−α (x−a)2(b−a)2−α f (a)∣∣∣∣∣ (8)≤ M (b− x)1−α(b−a)[(b−a)α (x−a)+(b− x)α (a+b−2x)2α]for α ≥ 1 and a≤ x < b.232 MEHMET ZEKI SARIKAYA - HATICE YALDIZ - NAGIHAN BASAKProof. From Lemma 2.1, we have∣∣∣∣∣12 f (x)− (α+1)Γ(α)(b− x)1−α2(b−a) Jαa f (b)+ 12Jα−1a (P2(x,b) f (b))+(b− x)2−α2(b−a) Γ(α)Jα−1a f (b)+(b− x)1−α (x−a)2(b−a)2−α f (a)∣∣∣∣∣=1Γ(α)∣∣∣∣∫ ba (b− t)α−1 K1 (x, t) f ′ (t)dt∣∣∣∣ .Taking into account the assumptions on the function f , it yields∣∣∣∣∣12 f (x)− (α+1)Γ(α)(b− x)1−α2(b−a) Jαa f (b)+ 12Jα−1a (P2(x,b) f (b))+(b− x)2−α2(b−a) Γ(α)Jα−1a f (b)+(b− x)1−α (x−a)2(b−a)2−α f (a)∣∣∣∣∣≤ M (b− x)1−α(b−a) x∫a(b− t)α−1∣∣∣∣t− a+ x2∣∣∣∣dt (9)+b∫x(b− t)α−1∣∣∣∣t− b+ x2∣∣∣∣dt .Noting the left hand side of (9) by J then integrating by parts the right hand sideof (9), we obtainJ ≤ M (b− x)1−α2(b−a)[(b−a)α − (b− x)αα(x−a)+ (b− x)α+1α].ConsequentlyJ ≤ M (b− x)1−α(b−a)[(b−a)α (x−a)+(b− x)α (a+b−2x)2α].This completes the proof.Remark 2.4. Letting α = 1, formula (8) reduces the following inequality∣∣∣∣∣∣12 f (x)− (x−b) f (b)− (x−a) f (a)2(b−a) − 1(b−a)b∫af (t)dt∣∣∣∣∣∣≤ M(b−a)[(x−a)2+(b− x)22].NEW FRACTIONAL INEQUALITIES 233which connected with Ostrowski-Gru¨ss type integral inequality. If we take x =a+b2 in this inequality, it follows that∣∣∣∣∣∣12 f(a+b2)+f (b)+ f (a)4− 1(b−a)b∫af (t)dt∣∣∣∣∣∣≤ M (b−a)4 .REFERENCES[1] X. L. Cheng, Improvement of some Ostrowski-Gru¨ss type inequalities, ComputersMath. Applic, 42 (2001), 109–114.[2] S. S. Dragomir - S. Wang, An inequality of Ostrowski-Gru¨ss type and its appli-cations to the estimation of error bounds for some special means and for homenumerical quadrature rules, Computers Math. Applic 33 (11) (1997), 15–20.[3] P. Cerone - S. S. Dragomir, Trapezoidal type rules from an inequalities point ofview, Handbook of Analytic-Computational Methods in Applied Mathematics,CRC Press, N. Y. 2000.[4] D. S. Mitrinovic - J. Pecaric - A. M. Fink, Inequalities for functions and theirintegrals and derivatives, Kluwer Academic, Dordrecht, 1994.[5] D. S. Mitrinovic - J. Pecaric - A. M. Fink, Classical and new inequalities in anal-ysis, Kluwer Academic, Dordrecht, 1993.[6] M. Matic - J. Pecaric - N. Ujevic, Improvement and further generalization of in-equalities of Ostrowski-Gru¨ss type,Computers Math. Appl. 39 (3-4) (2000), 161–175.[7] G. Anastassiou - M. R. Hooshmandasl - A. Ghasemi - F. Moftakharzadeh, Mont-gomery identities for fractional integrals and related fractional inequalities, J.Inequal. in Pure and Appl. Math. 10 (4) (2009), Art. 97, 6 pp.[8] S. Belarbi - Z. Dahmani, On some new fractional integral inequalities, J. Inequal.in Pure and Appl. Math 10 (3) (2009), Art. 97, 6 pp.[9] Z. Dahmani - L. Tabharit - S. Taf, Some fractional integral inequalities, NonlinearScience Letters A 2 (1) (2010), 155–160.[10] Z. Dahmani - L. Tabharit - S. Taf, New inequalities via Riemann-Liouville frac-tional integration, J. Advance Research Sci. Comput. 2 (1) (2010), 40–45.[11] R. Gorenflo - F. Mainardi, Fractional calculus: integral and differentiable equa-tions of fractional order, Springer Verlag, Wien, 223-276, 1997.[12] D. S. Mitrinovic - J. E. Pecaric - A. M. Fink, Inequalities involving functions andtheir integrals and derivatives, Kluwer Academic Publishers, Dordrecht, 1991.[13] S. S. Dragomir - N. S. Barnett, An Ostrowski type inequality for mappings whose234 MEHMET ZEKI SARIKAYA - HATICE YALDIZ - NAGIHAN BASAKsecond derivatives are bounded and applications, RGMIA Research Report Col-lection V.U.T. 1 (1999), 67–76.[14] S. S. Dragomir, An Ostrowski type inequality for convex functions, Univ. Beograd.Publ. Elektrotehn. Fak. Ser. Mat. 16 (2005), 12–25.[15] Z. Liu, Some companions of an Ostrowski type inequality and application, J. In-equal. in Pure and Appl. Math 10 (2) (2009), Art. 52, 12 pp.[16] S. G. Samko - A. A Kilbas - O. I. Marichev, Fractional integrals and derivatives.Theory and application, Gordan and Breach Science, New York, 1993.[17] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ.Comenianae LXXIX 1 (2010), 129–134.[18] M. Z. Sarikaya, On the Ostrowski type integral inequality for double integrals,Demonstratio Mathematica 45 3 (2012), 533–540.[19] M. Z. Sarikaya - H. Ogunmez, On the weighted Ostrowski type integral inequalityfor double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics 36 (2011), 1153–1160.[20] M. Z. Sarikaya - H. Ogunmez, On new inequalities via Riemann-Liouville frac-tional integration, Abstract and Applied Analysis 2012 (2012), Article ID 428983,10 pages.[21] M. Z. Sarikaya - H. Yaldiz, On weighted Montogomery identities for Riemann-Liouville fractional integrals, Konuralp J. of Mathematics 1 (1) (2013), 48–53.[22] M. Z. Sarikaya - E. Set - H. Yaldiz - N. Basak, Hermite-Hadamard’s inequali-ties for fractional integrals and related fractional inequalities, Mathematical andComputer Modelling 57 (2013), 2403–2407.[23] A. Rafiq - F. Ahmad, Another weighted Ostrowski-Gru¨ss type inequality for twicedifferentiable mappings, Kragujevac Journal of Mathematics 31 (2008), 43–51.[24] F. Tong - L. Guan, A simple proof of the generalized Ostrowski-Gru¨ss type integralinequality, Int. Jour. of Math. Analysis 2 (18) (2008), 889–892.MEHMET ZEKI SARIKAYADepartment of MathematicsFaculty of Science and ArtsDu¨zce University, Du¨zce, Turkeye-mail: sarikayamz@gmail.comHATICE YALDIZDepartment of MathematicsFaculty of Science and ArtsDu¨zce University, Du¨zce, Turkeye-mail: yaldizhatice@gmail.comNEW FRACTIONAL INEQUALITIES 235NAGIHAN BASAKDepartment of MathematicsFaculty of Science and ArtsDu¨zce University, Du¨zce, Turkeye-mail: nagihan.basak@hotmail.com

New fractional inequalities of Ostrowski-Grüss type

Mehmet Zeki Sarikaya

Hatice Yaldiz

Nagihan Basak

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New fractional inequalities of Ostrowski-Grüss type

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