Akdemir, Ahmet Ocak

Mumcu, Ilker

Set, Erhan

English

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 20 (2019), No. 2, pp. 1227–1236 DOI: 10.18514/MMN.2019.2766CHEBYSHEV TYPE INEQUALITIES FOR CONFORMABLEFRACTIONAL INTEGRALSERHAN SET, AHMET OCAK AKDEMIR, AND I˙LKER MUMCUReceived 29 November, 2018Abstract. In this article, firstly some necessary definitions and results involving fractional integ-rals are given. Secondly, a new identity involving conformable fractional integrals is given. Then,by using this identity, we establish new Chebyshev inequalities for the Chebyshev functional viaconformable fractional integral.2010 Mathematics Subject Classification: 26A33; 26D10; 26D15; 33B15.Keywords: Chebyshev inequality, Riemann-Liouville fractional integral, conformable fractionalintegrals1. INTRODUCTIONThere are many different type integral inequalities in the literature such as classicaland analytic inequalities. We will start with the most prominent one of classicalinequalities that is called Chebyshev inequality.This inequality was established by Chebyshev in [3] as following;jT .f;g/j  112.b a/2f 01g01 ; (1.1)where f;g W Œa;b!R are absolutely continuous functions whose derivatives f 0;g0 2L1 Œa;b andT .f;g/D 1b abZaf .x/g .x/dx 0@ 1b abZaf .x/dx1A0@ 1b abZag .x/dx1A ;(1.2)which is called the Cˇebysˇev functional, provided the integrals in (1.2) exist.We will remind some well-known concepts as followings:The beta function defined as follows [5, p18]:B .a;b/D   .a/  .b/  .aCb/ DZ 10ta 1 .1  t /b 1dt; a;b > 0;where   .˛/D R10 e tu˛ 1du is Gamma function.c 2019 Miskolc University Press1228 ERHAN SET, AHMET OCAK AKDEMIR, AND I˙LKER MUMCUDefinition 1. Let f 2L1Œa;b. The Riemann-Liouville integrals J ˛aCf and J ˛b fof order ˛ > 0 are defined byJ ˛aCf .x/D1  .˛/Z xa.x  t /˛ 1f .t/dt; x > aandJ ˛b f .x/D1  .˛/Z bx.t  x/˛ 1f .t/dt; x < brespectively. Here J 0aCf .x/D J 0b f .x/D f .x/.In [2], Belarbi and Dahmani established following theorems for the Chebyshevinequalities.Theorem 1. Let f and g be two synchronous functions on Œ0;1/. Then for allt > 0; ˛ > 0, we have:J ˛.fg/   .˛C1/t˛J ˛f .t/J ˛g.t/: (1.3)Theorem 2. Let f and g be two synchronous functions on Œ0;1/. Then for allt > 0; ˛ > 0; ˇ > 0, we have:t˛  .˛C1/Jˇ .fg/.t/C tˇ  .ˇC1/J˛.fg/.t/ J ˛f .t/J ˇg.t/CJ ˇf .t/J ˛g.t/(1.4)Theorem 3. Let .fi /iD1;:::;n be n positive inreasing functions on Œ0;1/. Then forany t > 0; ˛ > 0, we haveJ ˛ nYiD1fi!.t/ .J ˛.1//1 nnYiD1J ˛fi .t:/ (1.5)Theorem 4. Let f and g be two functions defined on Œ0;1/, such that f is in-crerasing, g is differentiable and there exist a real number m WD inft0g0.t/. Thenthe inequalityJ ˛.fg/.t/ .J ˛.1// 1J ˛f .t/J ˛g.t/  mt˛C1J˛f .t/CmJ ˛.tf .t// (1.6)is valid for all t > 0; ˛ > 0.In [4], Khalil et al. define a new well-behaved simple fractional derivative calledconformable fractional derivative depending just on the basic limit definition of thederivative. They also defined the fractional integral of order 0 < ˛  1 only.In [1], Abdeljawad gave the definition of left and right conformable fractionalintegrals of any order ˛ > 0.CHEBYSHEV TYPE INEQUALITIES FOR CONFORMABLE FRACTIONAL INTEGRALS 1229Definition 2. Let ˛ 2 .n;nC1 and set ˇ D ˛ n then the left conformable frac-tional integral starting at a if order ˛ is defined by.I a˛ f /.t/D1nŠZ ta.t  x/n.x a/ˇ 1f .x/dxAnalogously, the right conformable fractional integral is defined by.bI˛f /.t/D 1nŠZ bt.x  t /n.b x/ˇ 1f .x/dx:In [7], Set and Mumcu gave following LemmaLemma 1. If g W Œa;b! R is integrable and symmetric to .aCb/=2 with a < b,thenI a˛ g.b/D bI˛g.a/D12ŒbI˛g.a/CI a˛ g.b/with ˛ 2 .n;nC1.Notice that if ˛ D nC 1 then ˇ D ˛ nD nC 1 nD 1 and hence .I a˛ f /.t/D.J anC1f /.t/.For some recent results related to conformable fractional integrals, see, e.g., [6, 7].The main purpose of this paper is to establish Chebyshev type inequalities for con-formable fractional integral.2. CHEBYSHEV TYPE INEQUALITIESLemma 2. Let f W Œa;b! RC be function, where 0  a < b: If f 2 LŒa;b thenwe getZ 10tn.1  t /˛ n 1f .taC .1  t /b/dtCZ 10tn.1  t /˛ n 1f ..1  t /aC tb/dtD nŠ.b a/˛ ŒbI˛f .a/CI a˛ f .b/:for ˛ 2 .n;nC1; nD 0;1;2; :::.Proof. By integration by parts, we can stateZ 10tn.1  t /˛ n 1f .taC .1  t /b/dt DD 1b aZ bab xb anx ab a˛ n 1f .x/dxD 1.b a/˛Z ba.b x/n.x a/˛ n 1f .x/dxD nŠ.b a/˛ .Ia˛ f /.b/ (2.1)1230 ERHAN SET, AHMET OCAK AKDEMIR, AND I˙LKER MUMCUand Z 10tn.1  t /˛ n 1f ..1  t /aC tb/dt DD 1b aZ bax ab anb xb a˛ n 1f .x/dxD 1.b a/˛Z ba.x a/n.b x/˛ n 1f .x/dxD nŠ.b a/˛ .bI˛f /.a/: (2.2)Adding (2.1) and (2.2), we get desired result. Theorem 5. Let f;g W Œa;b! R are two monotonic functions of the same mono-tonicity, f;g 2LŒa;b and ˛ 2 .n;nC1 .nD 0;1;2; :::/, then the following inequal-ity holds:  .˛C1/  .˛ n/2.b a/˛ /ŒbI˛.fg/.a/CI a˛ .fg/.b/   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/  .˛C1/2.b a/˛ ŒbI˛g.a/CI a˛ g.b/: (2.3)Proof. Our proof is based on the property of positivity of the integral. The basicremark is the inequality.f .x/ f .y//.g.x/ g.y// 0for all x;y 2 Œa;b. For x D taC .1  t /b, t 2 Œ0;1 we obtain.f .taC .1  t /b/ f .y//.g.taC .1  t /b/ g.y// 0: (2.4)Similarly, for x D .1  t /aC tb, t 2 Œ0;1 we have.f ..1  t /aC tb/ f .y//.g..1  t /aC tb/ g.y// 0: (2.5)Adding the inequalities (2.4) and (2.5) , then multiplying both sides with tn.1 t /˛ n 1 and integrating with respect to t over Œ0;1 we have:Z 10tn.1  t /˛ n 1.fg/.taC .1  t /b/dtCZ 10tn.1  t /˛ n 1.fg/..1  t /aC tb/dtC2B.nC1;˛ n/f .y/g.y/ g.y/Z 10tn.1  t /˛ n 1f .taC .1  t /b/dtCZ 10tn.1  t /˛ n 1f ..1  t /aC tb/dtCHEBYSHEV TYPE INEQUALITIES FOR CONFORMABLE FRACTIONAL INTEGRALS 1231Cf .y/Z 10tn.1  t /˛ n 1g.taC .1  t /b/dtCZ 10tn.1  t /˛ n 1g..1  t /aC tb/dt: (2.6)We apply Lemma 2 and multiplying both sides by 1nŠ1.b a/˛ ŒbI˛.fg/.a/CI a˛ .fg/.b/C2nŠB.nC1;˛ n 1/f .y/g.y/ g.y/.b a/˛ ŒbI˛f .a/CI a˛ f .b/Cf .y/.b a/˛ ŒbI˛g.a/CI a˛ g.b/: (2.7)We put y D taC .1  t /b, t 2 Œ0;1 in (2.7), whence it follows1.b a/˛ ŒbI˛.fg/.a/CI a˛ .fg/.b/C2nŠB.nC1;˛ n 1/.fg/.taC .1  t /b/ g.taC .1  t /b/.b a/˛ ŒbI˛f .a/CI a˛ f .b/Cf .taC .1  t /b/.b a/˛ ŒbI˛g.a/CI a˛ g.b/:(2.8)On the other hand, for y D .1  t /aC tb, t 2 Œ0;1 in (2.7) we have1.b a/˛ ŒbI˛.fg/.a/CI a˛ .fg/.b/C2nŠB.nC1;˛ n 1/.fg/..1  t /aC tb/ g..1  t /aC tb/.b a/˛ ŒbI˛f .a/CI a˛ f .b/Cf ..1  t /aC tb.b a/˛ ŒbI˛g.a/CI a˛ g.b/:(2.9)Adding the inequalities (2.8) and (2.9), then the multiplying both sides tn.1 t /˛ n 1and integrating with respect to t over Œ0;1, one has2.b a/˛B.nC1;˛ n/ŒbI˛.fg/.a/CI a˛ .fg/.b/ (2.10)C 2nŠB.nC1;˛ n/Z 10tn.1  t /˛ n 1.fg/.taC .1  t /b/CZ 10tn.1  t /˛ n 1.fg/..1  t /aC tb/ 1.b a/˛ ŒbI˛f .a/CI a˛ f .b/Z 10tn.1  t /˛ n 1g.taC .1  t /b/Z 10tn.1  t /˛ n 1g..1  t /aC tb/C 1.b a/˛ ŒbI˛g.a/CI a˛ g.b/1232 ERHAN SET, AHMET OCAK AKDEMIR, AND I˙LKER MUMCUZ 10tn.1  t /˛ n 1f .taC .1  t /b/Z 10tn.1  t /˛ n 1f ..1  t /aC tb/:We apply Lemma 24.b a/˛B.nC1;˛ n/ŒbI˛.fg/.a/CI a˛ .fg/.b/ 2nŠ.b a/2˛ ŒbI˛f .a/CI a˛ f .b/ŒbI˛g.a/CI a˛ g.b/:Multiplying both sides of the last inequality with .  .˛C1//28and using the propertiesof the beta functions, we obtain desired inequality. Remark 1. If we take ˛ D nC 1 in above theorem, then the inequality (2.3) be-comes:  .˛C1/2.b a/˛ /ŒJ˛aC.fg/.b/CJ ˛b .fg/.a/   .˛C1/2.b a/˛ ŒJ˛aCf .b/CJ ˛b f .a/  .˛C1/2.b a/˛ ŒJ˛aCg.b/CJ ˛b g.a/: (2.11)Remark 2. In above theorem, if we take ˛ D nC 1, b D t , a D 0 and f; g issymmetric to .aCb/=2, and use Lemma 1 then inequality (2.3) becomes inequality(1.3).Theorem 6. Let f;g W Œa;b! R are two monotonic functions of the same mono-tonicity, f;g 2LŒa;b, ˛ 2 Œn;nC1.nD 0;1;2; :::/ and ˇ 2 .k;kC1.kD 0;1;2; :::/,then2.b a/ˇ B.kC1;ˇ k/ŒbI˛.fg/.a/CI a˛ .fg/.b/C 2.b a/˛B.nC1;˛ n/ŒbIˇ .fg/.a/CI aˇ .fg/.b/ 1.b a/˛Cˇ ŒbI˛f .a/CI a˛ f .b/ŒbIˇg.a/CI aˇ g.b/C 1.b a/˛Cˇ ŒbI˛g.a/CI a˛ g.b/ŒbIˇf .a/CI aˇ f .b/: (2.12)Proof. Adding the inequalities (2.8) and (2.9),and then multiplying both sides ofobtained inequality with tk.1  t /ˇ k 1 and integrating with respect to t over Œ0;1,we get2.b a/ˇ B.nC1;ˇ n 1/ŒbI˛.fg/.a/CI a˛ .fg/.b/ (2.13)C 2nŠB.nC1;˛ n 1/Z 10tk.1  t /ˇ k 1.fg/.taC .1  t /b/CHEBYSHEV TYPE INEQUALITIES FOR CONFORMABLE FRACTIONAL INTEGRALS 1233CZ 10tk.1  t /ˇ k 1.fg/..1  t /aC tb/ 1.b a/˛ ŒbI˛f .a/CI a˛ f .b/Z 10tk.1  t /ˇ k 1g.taC .1  t /b/Z 10tk.1  t /ˇ k 1g..1  t /aC tb/C 1.b a/˛ ŒbI˛g.a/CI a˛ g.b/Z 10tk.1  t /ˇ k 1f .taC .1  t /b/Z 10tk.1  t /ˇ k 1f ..1  t /aC tb/:So, if we use Lemma 2, we have desired result. Remark 3. Applying Theorem 6 for ˛ D ˇ, we obtain Theorem 5.Remark 4. If we take ˛D nC1 and ˇD kC1 in theorem 6, then inequality (2.12)becomes:2.b a/kC1.kC1/ ŒJ˛aCfg.b/CJ ˛b fg.a/C 2.b a/nC1.nC1/ ŒJˇaCfg.b/CJ ˇb fg.a/ 1.b a/nCkC2 ŒJ˛aCf .b/CJ ˛b f .a/ŒJ ˇaCg.b/CJ ˇb g.a/C 1.b a/nCkC2 ŒJ˛aCg.b/CJ ˛b g.a/ŒJ ˇaCf .b/CJ ˇb f .a/:Remark 5. In above theorem, if we take ˛ D nC 1, b D t , a D 0, and f; g issymmetric to .aCb/=2, and use Lemma 1, then inequality (2.12) becomes inequality(1.4).Theorem 7. Let .fi /iD1;:::;n be a positive increasing functions on [a,b]. For ˛ 2.n;nC1 nD 0;1;2; :::, we havebI˛ nYiD1fi!.a/CI a˛ nYiD1fi!.b/  .˛C1/2.b a/˛  .˛ n/n 1 nYiD1ŒbI˛.fi /.a/CI a˛ .fi /.b/: (2.14)Proof. We prove this theorem by induction. For nD 1, we haveŒbI˛f1.a/CI a˛ f1.b/ ŒbI˛f1.a/CI a˛ f1.b/:1234 ERHAN SET, AHMET OCAK AKDEMIR, AND I˙LKER MUMCUFor nD 2, applying Theorem 5, we getŒbI˛.f1f2/.a/CI a˛ .f1f2/.b/   .˛C1/2.b a/˛  .˛ n/ŒbI˛.f1/.a/CI a˛ .f1/.b/ŒbI˛f2/.a/CI a˛ .f2/.b/:Now suppose thatbI˛ n 1YiD1fi!.a/CI a˛ n 1YiD1fi!.b/  .˛C1/2.b a/˛  .˛ n/n 2 n 1YiD1ŒbI˛.fi /.a/CI a˛ .fi /.b/: (2.15)Since .fi /iD1;:::;n are positive increasing functions, thenQn 1iD1 fi.t/ is an increas-ing function. So we can apply Theorem 5 to the functionsQn 1iD1 fi D g;fn D f , weobtain bI˛ nYiD1fi!.a/CI a˛ nYiD1fi!.b/  .˛C1/2.b a/˛  .˛ n/bI˛ n 1YiD1fi!.a/CI a˛ n 1YiD1fi!.b/ŒbI˛.fn/.a/CI a˛ .fn/.b/:Using (2.15), we get desired result. Remark 6. If we take ˛ D nC1 in Theorem 7, then inequality (2.14) becomes:J ˛b  nYiD1fi!.a/CJ ˛aC nYiD1fi!.b/.nC1/Š2.b a/˛n 1 nYiD1ŒJ ˛aCfi .b/CJ ˛b fi .a/: (2.16)Remark 7. If we take ˛ D nC1, b D t , aD 0 and fi I i D 1; :::;n is symmetric to.aCb/=2, then, using Lemma 1, inequality (2.14) becomes inequality (1.5).Theorem 8. Let f and g be two functions defined on Œ0;1/, such that f is in-creasing, g is differentiable and there exist a real number m WD inft0g0.t/. Thenthe inequality  .˛ n/ŒbI˛.fg/.a/CI a˛ .fg/.b/ (2.17)CHEBYSHEV TYPE INEQUALITIES FOR CONFORMABLE FRACTIONAL INTEGRALS 1235   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/ŒbI˛g.a/CI a˛ g.b/    .nC2/  .ˇ n/2.b a/˛.˛C1/ ŒbI˛f .a/CI a˛ f .b/CmŒbI˛.tf /.a/CI a˛ .tf /.b/:is valid for all t > 0; ˛ 2 .n;nC1;nD 0;1;2; :::.Proof. We consider the function h.t/ WD g.t/ mt . It is clear that h is differenti-able and it is increasing on Œ0;1/. Then by using Theorem 5, we get  .˛ n/ŒbI˛..g mt/f /.a/CI a˛ ..g mt/f /.b/   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/ŒbI˛g.a/CI a˛ g.b/ mbI˛t .a/ mI a˛ t .b/   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/ŒbI˛g.a/CI a˛ g.b/ mB.nC2;˛ n/   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/ŒbI˛g.a/CI a˛ g.b/    .nC2/  .ˇ n/2.b a/˛.˛C1/ ŒbI˛f .a/CI a˛ f .b/:So  .˛ n/ŒbI˛.fg/.a/CI a˛ .fg/.b/   .˛C1/2.b a/˛ ŒbI˛f .a/CI a˛ f .b/ŒbI˛g.a/CI a˛ g.b/    .nC2/  .ˇ n/2.b a/˛.˛C1/ ŒbI˛f .a/CI a˛ f .b/CmŒbI˛.tf /.a/CI a˛ .tf /.b/:which the proof is completed. Remark 8. In Theorem 8, if we take ˛ D nC 1 and ˇ D kC 1, inequality (2.17)becomes:ŒJˇaC.fg/.b/CJ ˇb fg/.a/ .nC2/Š2.b a/˛ ŒJˇaCf .b/CJ ˇb f .a/ŒJ ˇaCg.b/CJ ˇb g.a/ (2.18)    .nC2/  .ˇ n/2.b a/˛.˛C1/ ŒJˇaCf .b/CJ ˇb f .a/CmŒJ ˇaCf .b/CJ ˇb g.a/:Remark 9. In Theorem 8, if we take ˛D nC1 and f; g is symmetric to .aCb/=2,then, using Lemma 1 inequality (2.18) becomes inequality (1.6).1236 ERHAN SET, AHMET OCAK AKDEMIR, AND I˙LKER MUMCUREFERENCES[1] T. Abdeljawad, “On conformable fractional calculus,” Journal of Computational and Applied Math-ematics, vol. 279, pp. 57–66, 2015, doi: 10.1016/j.cam.2014.10.016.[2] S. Belarbi and Z. Dahmani, “On some new fractional integral inequalities,” J. Ineq. Pure Appl.Math., vol. 10, no. 3, pp. Article 86, 5 pp., 2009.[3] P. Chebyshev, “Sur les expressions approximatives des integrales definies par les autres prises entreles memes limites,” Proc. Math. Soc. Charkov, vol. 2, pp. 93–98, 1882.[4] R. Khalil, M. A. Horani, A. Yousef, and M. Sababheh, “A new definition of fractional deriv-ative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70., 2014, doi:10.1016/j.cam.2014.01.002.[5] E. Rainville, Special Functions. The Mcmillan Company, New York, 1960.[6] E. Set, A. Akdemir, and I. Mumcu, “Hadamard’s inequality and its extensions for conformablefractional integrals of any order ˛ > 0,” Creat. Math. Inform., vol. 27, no. 2, pp. 197–206, 2018.[7] E. Set and I˙.Mumcu, “Hermite-hadamard-fejer type inequalities for conformable fractional integ-rals,” Miskolc Math. Notes, vol. 20, no. 1, pp. 475–488, 2019, doi: 10.18514/MMN.2019.2421.Authors’ addressesErhan SetOrdu University, Department of Mathematics, Faculty of Science and Arts, Ordu, TurkeyE-mail address: erhanset@yahoo.comAhmet Ocak AkdemirAgˇrı I˙brahim C¸ec¸en University, Department of Mathematics, Faculty of Science and Arts, Agˇrı,TurkeyE-mail address: ahmetakdemir@agri.edu.trI˙lker MumcuOrdu University, Department of Mathematics, Faculty of Science and Arts, Ordu, TurkeyE-mail address: mumcuilker@msn.com

Chebyshev type inequalities for conformable fractional integrals

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