On Fractional Integral Inequalities Involving Hypergeometric Operators

1 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey 2Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 3 Institute of Space Sciences, Magurele Bucharest, Romania 4Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur 313001, India 5 Department of Mathematics & Statistics, J.E.C.R.C. University, Jaipur 303905, India


Introduction
Fractional integral inequalities have many applications; the most useful ones are in establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations.Further, they also provide upper and lower bounds to the solutions of the above equations.These considerations have led various researchers in the field of integral inequalities to explore certain extensions and generalizations by involving fractional calculus operators.One may, for instance, refer to such type of works in the book [1] and the papers [2][3][4][5][6][7][8][9][10][11].
In a recent paper, Purohit and Raina [9] investigated certain Chebyshev type [12] integral inequalities involving the Saigo fractional integral operators and also established the -extensions of the main results.The aim of this paper is to establish certain generalized integral inequalities for synchronous functions that are related to the Chebyshev functional using the fractional hypergeometric operator, introduced by Curiel and Galu é [13].Results due to Purohit and Raina [9] and Belarbi and Dahmani [2] follow as special cases of our results.
In the sequel, we use the following definitions and related details. (in terms of the Gauss hypergeometric function) of order  for a real-valued continuous function () is defined by [13] (see also [14]): where the function 2  1 (−) appearing as a kernel for the operator (2) is the Gaussian hypergeometric function defined by and ()  is the Pochhammer symbol: The object of the present investigation is to obtain certain Chebyshev type integral inequalities involving the generalized fractional integral operators [13] which involves in the kernel, the Gauss hypergeometric function (defined above).The concluding section gives some special cases of the main results.

Main Results
Our results in this section are based on the preliminary assertions giving composition formula of fractional integral (2) with a power function.Lemma 4. Let , , ,  ∈ R,  > −1,  +  > 0, and  −  +  > 0, then the following image formula for the power function under the operator (2) holds true: Proof.To prove (5), we take () =  −1 in the definition of fractional integral operator  ,,,  (⋅), given by ( 2), the lefthand side (say L) yields to Now, on using the following integral formula involving the Gaussian hypergeometric function [15, page 106, equation (3.118)]: then (6) immediately leads to the result (5).Now, we obtain certain integral inequalities for the synchronous functions involving the generalized fractional integral operator (2).

Theorem 5. Let 𝑓 and 𝑔 be two synchronous functions on
Proof.Let  and  be two synchronous functions; then using Definition 1, for all ,  ∈ (0, ),  ≥ 0, we have which implies that Consider We observe that each term of the above series is positive in view of the conditions stated with Theorem 5, and hence the function (, ) remains positive, for all  ∈ (0, ) ( > 0).Multiplying both sides of (10) by (, ) (where (, ) is given by ( 11)) and integrating with respect to  from 0 to , and using (2), we get  Next, multiplying both sides of (12) by (, ) ( ∈ (0, ),  > 0), where (, ) is given by (11), and integrating with respect to  from 0 to , and using formula (5) (for  = 1), we arrive at the desired result (8).
The following results give a generalization of Theorem 5. Theorem 6.Let  and  be two synchronous functions on [0, ∞) then Proof.To prove the above theorem, we use inequality (12).Multiplying both sides of ( 12) by which remains positive in view of the conditions stated with (13) which on using (5) readily yields the desired result (13).
Proof.We prove this theorem by mathematical induction. Clearly which holds in view of (8) of Theorem 5.
By the induction principle, we suppose that the inequality holds true for some positive integer  ≥ 2. Now (  ) =1,..., are increasing functions which imply that the function ∏ −1 =1   () is also an increasing function.Therefore, we can apply inequality (8) Now, on making use of formula (5) (for  = 2), we are lead to the result (22) after some simplifications.
Proof.By applying the similar procedure as of Theorem 10, one can easily establish the above theorem.Therefore, we omitted the details of the proof of this theorem.

Special Cases
We now briefly consider some consequences of the results derived in the previous section.Following Curiel and Galu é [13], the operator (2) would reduce immediately to the extensively investigated Saigo, Erd é lyi-Kober, and Riemann-Liouville type fractional integral operators, respectively, given by the following relationships (see also [14,16] Now, if we consider  = 0 (and ] = 0 additionally for Theorem 6) and make use of relation (25), Theorems 5 to 9 provide, respectively, the known fractional integral inequalities due to Purohit and Raina [9].
Again, for  = 0 Theorems 10 and 11 provide, respectively, the following inequalities involving Saigo fractional integral operators.
We conclude with the remark that the results derived in this paper are general in character and give some contributions to the theory integral inequalities and fractional calculus.Moreover, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations.
and integrating with respect to  from 0 to , we get , for  = 1 in (17), we have