Abstract and Applied Analysis

Volume 2014 (2014), Article ID 609160, 10 pages

http://dx.doi.org/10.1155/2014/609160

## Chebyshev Type Integral Inequalities Involving the Fractional Hypergeometric Operators

^{1}Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia^{2}Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey^{3}Institute of Space Sciences, Magurele, Bucharest, Romania^{4}Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur 313001, India

Received 31 January 2014; Accepted 8 March 2014; Published 22 April 2014

Academic Editor: Juan J. Nieto

Copyright © 2014 D. Baleanu and S. D. Purohit. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By making use of the fractional hypergeometric operators, we establish certain new fractional integral inequalities for synchronous functions which are related to the weighted version of the Chebyshev functional. Some consequent results and special cases of the main results are also pointed out.

#### 1. Introduction

Fractional integral inequalities have proved to be one of the most important and powerful tools for the development of many branches of pure and applied mathematics. These inequalities have many applications in numerical quadrature, transform theory, probability, and statistical problems, but the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems. Moreover, they also provide upper and lower bounds to the solutions of the above equations. Therefore, a significant development in the classical and fractional integral inequalities, particularly in analysis, has been witnessed; see, for instance, the papers [1–7] and the references cited therein.

Let the functional where and are two integrable functions and synchronous on ; that is, for any ; then the Chebyshev integral inequality [8] is given by . The sign of this inequality is reversed if and are asynchronous on (i.e., , for any ). Under various assumptions (Chebyshev inequalities, Grüss inequality, etc.) [9–11], Chebyshev functionals are useful to give a lower bound or an upper bound for , in the theory of approximations. Therefore, in the literature, we found several papers that analyze extensions and generalizations of these integral inequalities by involving fractional calculus and -calculus operators. One may refer to such type of works in the book [12] and the recent papers [13–24]. A similar type of important role is also played by the Chebyshev polynomials of first and second kind in the theory of approximation. For a detailed account about these polynomials and their generating functions, one can see a recent paper [25] and research monograph by Srivastava and Manocha [26].

For the present paper, let us consider the weighted version of the Chebyshev functional (see [8]) provided that and are two integrable functions on and is a positive and integrable function on . In 2000, Dragomir [27] derived the following inequality: where are two differentiable functions and , , , . Recently, Dahmani et al. [28] established some integral results related to Chebyshev’s functional (3) in the case of differentiable functions whose derivatives belong to the space , involving Riemann-Liouville fractional integrals. Purohit and Raina [22, 29] added one more dimension to this study by introducing certain new integral inequalities for synchronous functions, involving the Saigo fractional integral operators [30]. Further, Baleanu et al. [24] established certain generalized integral inequalities for synchronous functions that are related to the Chebyshev functional (1) using the fractional hypergeometric operator, introduced by Curiel and Galué [31].

In this paper, we establish certain integral inequalities related to the weighted Chebyshev’s functional (3) in the case of differentiable functions whose derivatives belong to the space , involving fractional hypergeometric operators due to [31]. Later, we develop some integral inequalities for the fractional integrals by suitably choosing the function . Some of the known results due to Dahmani et al. [28] and Purohit and Raina [29] follow as special cases of our findings.

Firstly, we give some necessary definitions and mathematical preliminaries of fractional calculus operators which are used further in this paper.

*Definition 1. *A real-valued function () is said to be in the space (), if there exists a real number such that , where .

*Definition 2. *Let , , ; then, a generalized fractional integral (in terms of the Gauss hypergeometric function) of order for a real-valued continuous function is defined by [31] (see also [32])
where the function appearing as a kernel for the operator (5) is the Gaussian hypergeometric function defined by
and is the Pochhammer symbol
It may be noted that the Pochhammer symbol in terms of the gamma function is defined by
where the gamma function [33] is given by

Our results in this paper are based on the following preliminary assertions giving composition formula of fractional integral (5) with a power function.

Lemma 3 (see [24]). *Let , , , and ; then, the following image formula for the power function under the operator (5) holds true:
*

*2. Main Results *

*In this section, we obtain certain integral inequality which gives estimation for the fractional integral of a product in terms of the product of the individual function fractional integrals, involving fractional hypergeometric operators. We give our results related to Chebyshev’s functional (3) in the case of differentiable mappings whose derivatives belong to the space satisfying Holder’s inequality.*

*Theorem 4. Let be a positive function and let and be two synchronous functions on . If , , , , then (for all , , , , )
*

*Proof. *Let and be two synchronous functions; then, using Definition 1, for all , , we define
Consider
We observe that each term of the above series is positive in view of the conditions stated with Theorem 4, and, hence, the function remains positive, for all ().

Multiplying both sides of (12) by (where is given by (13)) and integrating with respect to from to and using (5), we get
Next, on multiplying both sides of (14) by , where is given by (13), and integrating with respect to from to , we can write
In view of (12), we have
Using the following Holder inequality for the double integral:
we obtain
Since
then (18) reduces to
It follows from (15) that
Applying again Holder’s inequality (17) on the right-hand side of (21), we get
In view of the fact that
we get
From (24), we obtain
Since , the above inequality yields
which in view of (15) gives
Making use of (26) and (27), the left-hand side of the inequality (11) follows.

To prove the right-hand side of the inequality (11), we observe that , , and, therefore,
Evidently from (26), we get
which completes the proof of Theorem 4.

*The following gives a generalization of Theorem 4.*

*Theorem 5. Let be a positive function and let and be two synchronous functions on . If , , , , then
for all , , , , , , , , .*

*Proof. *To prove the above theorem, we use inequality (14). Multiplying both sides of (14) by
which remains positive in view of the conditions stated with (30) and then integrating with respect to from to , we get
Now making use of (20), then (32) gives
Applying Holder’s inequality (17) on the right-hand side of (33), we get
which on using (23) readily yields the following inequality:
In view of (32) and (35) and the properties of modulus, one can easily arrive at the left-sided inequality of Theorem 5. Moreover, we have , ; hence,
Therefore, from (35), we get
which completes the proof of Theorem 5.

*Remark 6. *For , , , and , Theorem 5 immediately reduces to Theorem 4.

*3. Consequent Results and Special Cases *

*3. Consequent Results and Special Cases*

*As implications of our main results, we consider some consequent results of Theorems 4 and 5 by suitably choosing the function . To this end, let us set ; then, on using (10), Theorems 4 and 5 yield the following results.*

*Corollary 7. Let and be two synchronous functions on . If , , , , then
where , , , , , , .*

*Corollary 8. Let and be two synchronous functions on . If , , , , then
for all , , , , , , , , , , .*

*Further, if we put in Corollaries 7 and 8 (or in Theorems 4 and 5), we obtain the following integral inequalities.*

*Corollary 9. Let and be two synchronous functions on . If , , , , then
where , , , , .*

*Corollary 10. Let and be two synchronous functions on . If , , , , then
for all , , , , , , , , .*

*We now, briefly consider some consequences of the results derived in the previous section. Following Curiel and Galué [31], the operator (5) would reduce immediately to the extensively investigated Saigo, Erdélyi-Kober, and Riemann-Liouville type fractional integral operators, respectively, given by the following relationships (see also [30, 32]):
*

*Now, if we consider (and additionally for Theorem 5) and make use of the relation (42), Theorems 4 and 5 provide, respectively, the known fractional integral inequalities due to Purohit and Raina [29]. Again, if we set and replace by (and and replace by additionally for Theorem 5) and make use of the relation (44), then Theorems 4 and 5 correspond to the known integral inequalities due to Dahmani et al. [28, pages 39–42, Theorems 3.1 to 3.2] involving the Riemann-Liouville type fractional integral operator.*

*Lastly, we conclude this paper by remarking that we have introduced new general extensions of Chebyshev type inequalities involving fractional integral hypergeometric operators. By suitably specializing the arbitrary function , one can further easily obtain additional integral inequalities involving the Riemann-Liouville, Erdélyi-Kober, and Saigo type fractional integral operators from our main results in Theorems 4 and 5.*

*Conflict of Interests*

*Conflict of Interests*

*The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgment*

*Acknowledgment*

*The authors would like to express their thanks to the referees of the paper for the various suggestions given for the improvement of the paper.*

*References*

*References*

- V. Lakshmikantham and A. S. Vatsala, “Theory of fractional differential inequalities and applications,”
*Communications in Applied Analysis*, vol. 11, no. 3-4, pp. 395–402, 2007. View at Google Scholar · View at MathSciNet - J. D. Ramírez and A. S. Vatsala, “Monotone iterative technique for fractional differential equations with periodic boundary conditions,”
*Opuscula Mathematica*, vol. 29, no. 3, pp. 289–304, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Denton and A. S. Vatsala, “Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations,”
*Opuscula Mathematica*, vol. 31, no. 3, pp. 327–339, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - A. Debbouche, D. Baleanu, and R. P. Agarwal, “Nonlocal nonlinear integrodifferential equations of fractional orders,”
*Boundary Value Problems*, vol. 2012, article 78, 10 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H.-R. Sun, Y.-N. Li, J. J. Nieto, and Q. Tang, “Existence of solutions for Sturm-Liouville boundary value problem of impulsive differential equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 707163, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - Z.-H. Zhao, Y.-K. Chang, and J. J. Nieto, “Asymptotic behavior of solutions to abstract stochastic fractional partial integrodifferential equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 138068, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Liu, J. J. Nieto, and Otero-Zarraquiños, “Existence results for a coupled system of nonlinear singular fractional differential equations with impulse effects,”
*Mathematical Problems in Engineering*, Article ID 498781, 21 pages, 2013. View at Google Scholar · View at MathSciNet - P. L. Chebyshev, “Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites,”
*Proceedings of the Mathematical Society of Kharkov*, vol. 2, pp. 93–98, 1882. View at Google Scholar - P. Cerone and S. S. Dragomir, “New upper and lower bounds for the Čebyšev functional,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 3, no. 5, article 77, 2002. View at Google Scholar · View at MathSciNet - G. Grüss, “Über das Maximum des absoluten Betrages von $(1/(b-a))\underset{a}{\overset{b}{\int}}f\left(x\right)g\left(x\right)dx-(1/{(b-a)}^{2})\underset{a}{\overset{b}{\int}}f\left(x\right)dx\underset{a}{\overset{b}{\int}}g\left(x\right)dx$,”
*Mathematische Zeitschrift*, vol. 39, no. 1, pp. 215–226, 1935. View at Publisher · View at Google Scholar · View at MathSciNet - D. S. Mitrinović, J. E. Pečarić, and A. M. Fink,
*Classical and New Inequalities in Analysis*, vol. 61, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1993. View at MathSciNet - G. A. Anastassiou,
*Advances on Fractional Inequalities*, Springer Briefs in Mathematics, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - J. Pečarić and I. Perić, “Identities for the Chebyshev functional involving derivatives of arbitrary order and applications,”
*Journal of Mathematical Analysis and Applications*, vol. 313, no. 2, pp. 475–483, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S. Belarbi and Z. Dahmani, “On some new fractional integral inequalities,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 10, no. 3, article 86, 2009. View at Google Scholar · View at MathSciNet - Z. Dahmani and L. Tabharit, “Certain inequalities involving fractional integrals,”
*Journal of Advanced Research in Scientific Computing*, vol. 2, no. 1, pp. 55–60, 2010. View at Google Scholar · View at MathSciNet - S. L. Kalla and A. Rao, “On Grüss type inequality for a hypergeometric fractional integral,”
*Le Matematiche*, vol. 66, no. 1, pp. 57–64, 2011. View at Google Scholar · View at MathSciNet - H. Öğünmez and U. M. Özkan, “Fractional quantum integral inequalities,”
*Journal of Inequalities and Applications*, vol. 2011, Article ID 787939, 7 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - W. T. Sulaiman, “Some new fractional integral inequalities,”
*Journal of Mathematical Analysis*, vol. 2, no. 2, pp. 23–28, 2011. View at Google Scholar · View at MathSciNet - M. Z. Sarikaya and H. Ogunmez, “On new inequalities via Riemann-Liouville fractional integration,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 428983, 10 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Başak, “Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities,”
*Mathematical and Computer Modelling*, vol. 57, no. 9-10, pp. 2403–2407, 2013. View at Publisher · View at Google Scholar · View at Scopus - Z. Dahmani and A. Benzidane, “On a class of fractional q-integral inequalities,”
*Malaya Journal of Matematik*, vol. 3, no. 1, pp. 1–6, 2013. View at Google Scholar - S. D. Purohit and R. K. Raina, “Chebyshev type inequalities for the Saigo fractional integrals and their $q$-analogues,”
*Journal of Mathematical Inequalities*, vol. 7, no. 2, pp. 239–249, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. D. Purohit, F. Uçar, and R. K. Yadav, “On fractional integral inequalities and their $q$-analogues,”
*Revista Tecnocientifica URU*. In press. - D. Baleanu, S. D. Purohit, and P. Agarwal, “On fractional integral inequalities involving hypergeometric operators,”
*Chinese Journal of Mathematics*, vol. 2014, Article ID 609476, 5 pages, 2014. View at Publisher · View at Google Scholar - C. Cesarano, “Identities and generating functions on Chebyshev polynomials,”
*Georgian Mathematical Journal*, vol. 19, no. 3, pp. 427–440, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. M. Srivastava and H. L. Manocha,
*A Treatise on Generating Functions*, Eliis Horwood Series: Mathematical and Its Applications, Ellis Horwood Ltd., Chichester, UK, 1984. View at MathSciNet - S. S. Dragomir, “Some integral inequalities of Grüss type,”
*Indian Journal of Pure and Applied Mathematics*, vol. 31, no. 4, pp. 397–415, 2000. View at Google Scholar · View at MathSciNet - Z. Dahmani, O. Mechouar, and S. Brahami, “Certain inequalities related to the Chebyshev's functional involving a Riemann-Liouville operator,”
*Bulletin of Mathematical Analysis and Applications*, vol. 3, no. 4, pp. 38–44, 2011. View at Google Scholar · View at MathSciNet - S. D. Purohit and R. K. Raina, “Certain fractional integral inequalities involving the Gauss hypergeometric function”.
- M. Saigo, “A remark on integral operators involving the Gauss hypergeometric functions,”
*Mathematical Reports of College of General Education, Kyushu University*, vol. 11, no. 2, pp. 135–143, 1977/78. View at Google Scholar · View at MathSciNet - L. Curiel and L. Galué, “A generalization of the integral operators involving the Gauss' hypergeometric function,”
*Revista Técnica de la Facultad de Ingeniería Universidad del Zulia*, vol. 19, no. 1, pp. 17–22, 1996. View at Google Scholar · View at MathSciNet - V. Kiryakova,
*Generalized Fractional Calculus and Applications*, vol. 301 of*Pitman Research Notes in Mathematics Series*, Longman Scientific & Technical, Harlow, UK, 1994. View at MathSciNet - G. E. Andrews, R. Askey, and R. Roy,
*Special Functions*, vol. 71 of*Encyclopedia of Mathematics and Its Applications*, Cambridge University Press, Cambridge, UK, 1999. View at MathSciNet

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