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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 249091, 7 pages
Certain Chebyshev Type Integral Inequalities Involving Hadamard’s Fractional Operators
1Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
2Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur 313001, India
3Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
Received 1 April 2014; Accepted 28 April 2014; Published 11 May 2014
Academic Editor: J. C. Prajapati
Copyright © 2014 Sotiris K. Ntouyas et al. 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.
We establish certain new fractional integral inequalities for the differentiable functions whose derivatives belong to the space , related to the weighted version of the Chebyshev functional, involving Hadamard’s fractional integral operators. As an application, particular results have been also established.
In 1882, Chebyshev  gave the following inequality.
If are absolutely continuous functions, whose first derivativesandare bounded and then where denotes the norm in .
The Chebyshev functional (1) has many applications in numerical quadrature, transform theory, probability, study of existence for solutions of differential equations, and in statistical problems. Moreover, in the theory of approximations, under various assumptions (Chebyshev inequalities, Grüss inequality, etc.), Chebyshev functionals are useful to give lower bound or upper bounds for the functions. Therefore, in the literature, we found several extensions and generalizations of these classical integral inequalities, including fractional calculus and -calculus operators also (see [2–14]).
Our work in the present paper is based on a weighted version of the Chebyshev functional (see ): where and are two integrable functions on and is a positive and integrable function on . In 2000, Dragomir  derived the following inequality: where and are two differentiable functions and , , , and . Recently, Dahmani et al.  added one more dimension to this study by introducing generalization of inequality (4), involving Riemann-Liouville fractional integrals. Moreover, Purohit and Raina [17–19] and Baleanu et al. [20, 21] introduced certain generalized integral inequalities for synchronous functions, involving the various fractional hypergeometric integral operators, while Tariboon et al.  studied Riemann-Liouville fractional integral inequalities.
In 1892, Hadamard  introduced a fractional derivative, which differs from the Riemann-Liouville and Caputo derivatives in the sense that the kernel of the integral contains logarithmic function of arbitrary exponent. For details and fundamental properties of Hadamard fractional derivative and integral can be found in [24–28]. Recently, some results on fractional integral inequalities have been derived by using Hadamard fractional integrals (see [29–32]).
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 Hadamard fractional integral operators . We also develop some integral inequalities for the fractional integrals by suitably choosing the function , as special cases of our findings.
Firstly, we mention below the basic definitions and notations of some well-known operators of fractional calculus, which shall be used in the sequel.
The Hadamard fractional integral of order of a function , for all , is defined as  Further, the Hadamard fractional derivative of order , , of a function is given by Our results in this paper are based on the following preliminary assertions giving composition formula of Hadamard fractional integral and derivatives with a power function () where .
2. Main Results
Our results in this section are related to the Chebyshev’s functional (3) in the case of differentiable mappings whose derivatives belong to the space and satisfying Holder’s inequality. Here, we obtain certain new integral inequalities which give an estimation for the fractional integral of a product in terms of the product of the individual function fractional integrals, involving Hadamard fractional integral operators.
Theorem 1. Let be a positive function and let and be two differentiable functions on. If , , , then, for all and ,
Proof. We define
We observe that the function remains positive, for all . Multiplying both sides of (9) by and integrating with respect to from to , we get
Next, multiplying both sides of (11) by and integrating with respect to from to , we can write
In view of (9), we have
Using the following Hölder’s inequality for and ,
therefore, inequality (15) reduces to
It follows from (12) that
Applying again Hölder’s inequality on the right-hand side of (18), we get
In view of the fact that
From (21), we obtain
Using the relation , the above inequality yields to
On the other hand, (12) gives
On making use of (23) and (24), the left-hand side of inequality (8) follows very easily.
Now, to prove the right-hand side of inequality (8), we observe that , , and therefore, Evidently, from (23), we get which completes the proof of Theorem 1.
Now, we establish the following integral inequality, which may be regarded as a generalization of Theorem 1.
Theorem 2. Let be a positive function and let and be two differentiable functions on . If , , , , then for all , , and .
Proof. To prove the above theorem, we use inequality (11). Multiplying both sides of (11) by
which remains positive and integrating with respect to from to , we get
Now making use of (17), (29) gives
Applying Hölder’s inequality on the right-hand side of (30), we get
In view of (29) and (32) and the properties of modulus, one can easily arrive at the left-sided inequality of Theorem 2. Moreover, we have , ; hence,
Therefore, from (32), we get
which completes the proof of Theorem 2.
3. Special Cases
As implications of our main results, we consider some consequent results of Theorems 1 and 2 by suitably choosing the function . Other classes of no weighted inequalities are also obtained. To this end, let us set ; then Theorems 1 and 2 yield the following results.
Corollary 4. Let and be two differentiable functions on . If , , , , then, for all , and ,
Corollary 5. Let and be two differentiable functions on . If , , , , then for all , , , and .
Corollary 6. Let and be two differentiable functions on . If , , , , then, for all and ,
Corollary 7. Let and be two differentiable functions on . If , , , , then for all , , and .
We conclude our paper by remarking that we have introduced new general Chebyshev type inequalities involving Hadamard fractional integral operators. By suitably specializing the arbitrary function, one can further easily obtain additional fractional integral inequalities from our main results (Theorems 1 and 2).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research of J. Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics- (NAAM-) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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