SEVERAL INTEGRAL INEQUALITIES FOR GENERALIZED RIEMANN-LIOUVILLE FRACTIONAL OPERATORS

In this paper, using a generalized integral operator, of the RiemannLiouville type, dened and studied in a previous work by the authors, we obtain various integral inequalities for positive functions, which contains several reported in the literature. Various remarks carried out throughout the work and pointed out in the Conclusions, show the scope and strength of our results, in particular, it is shown that under particular cases of the considered kernel, several known fractional integral operators are obtained. 1. INTRODUCTION Calculus, using di¤erent notions of derivatives and integrals of arbitrary order, has become in recent years one of the centers of attention of mathematical researchers, both pure and applied. By other hand, one of the most developed mathematical areas in the last 20 years is that of Integral Inequalities, associated with di¤erent functional notions: convex, synchronous functions within the framework of Riemann, fractional and generalized integral operators. Throughout the work we use the functions (see [8,9,10,11])and k (cf. dened by [2]): 2020 Mathematics Subject Classication. Primary 26A33; Secondary 26D10, 47A63.


INTRODUCTION
Calculus, using di¤erent notions of derivatives and integrals of arbitrary order, has become in recent years one of the centers of attention of mathematical researchers, both pure and applied.
By other hand, one of the most developed mathematical areas in the last 20 years is that of Integral Inequalities, associated with di¤erent functional notions: convex, synchronous functions within the framework of Riemann, fractional and generalized integral operators.
Throughout the work we use the functions (see [8,9,10,11])and k (cf. de…ned by [2]): It is clear that if k ! 1 we have k (z) ! (z), k (z) = (k) z k 1 z k and k (z + k) = z k (z). As well, we de…ne the k-beta function as follows u+v) . In [3] the following fractional integral operator of the Riemann-Liouville type is de…ned and its main properties are studied. De…nition 1. The k-generalized fractional Riemann-Liouville integral of order with 2 R, and s 6 = 1 of an integrable function (u) on [0; 1), are given as follows: In this work, it was shown that many of the known integral operators can be obtained as particular cases of them.
The main purpose of this paper, using the generalized fractional integral operator of the Riemann-Liouville type, from De…nition 1, is to establish several integral inequalities, which contain as particular cases, several of those reported in the literature.

MAIN RESULTS
Below we present several integral inequalities, in the framework of the operators of De…nition 1, the …rst of them is the following.
Theorem 2. Let ' be a positive non-decreasing continuous function on [a 1 ; a 2 ] and let h : [a 1 ; a 2 ] ! R + be a positive continuous function. Then for a 1 < a 2 ; > 0; > 0; > 0 and s 6 = 1, we have Proof. Since the function ' is positive continuous and non-decreasing on [a 1 ; a 2 ] then, for all > 0, > 0, u; v 2 [a 1 ; ], with v u, and a 1 < a 2 , we have Multiplying both sides of (6) by k k ( )[F( ;u)] 1 k , then we integrate the resulting inequality with respect to u over (a 1 ; ), we get Now, multiplying both sides of (7) by then we integrate the resulting inequality with respect to v over (a 1 ; ), it holds that which implies (4).
Remark 3. If we take the kernel F ( ; s) = s , with k > 0 and s 2 R, s 6 = 1, the previous theorem becomes the Theorem 6 of [5].
In the following result, two functional parameters are used, extending the previous theorem.
Proof. Multiplying both sides of (7) by , then we integrate the resulting inequality with respect to v over (a 1 ; ), we obtain (9).
Remark 5. If in Theorem 2 we take = , we obtain Theorem 4.

Remark 6.
Under the same conditions as the previous Remark, this result covers Theorem 7 of [5].
The following result generalizes the Theorem 2, by including an appropriate h function. Theorem 7. Let ' and be two positive continuous function on [a 1 ; a 2 ] such that ' is non-decreasing, is non-increasing, and let h : [a 1 ; a 2 ] ! R + be a positive continuous function. Then for a 1 < a 2 ; > 0; > 0; > 0 and s 6 = 1, we have Proof. Since the functions ' and are positive continuous on [a 1 ; a 2 ] with ' is non-decreasing and is non-increasing, then for all > 0, > 0, u; v 2 [a 1 ; ], with v u, and a 1 < a 2 , we have Multiplying (11) by 1 k , then we integrate the resulting inequality with respect to u over (a 1 ; ), it holds that Now, we multiple (12) by (10).
Theorem 9. Let ' and be two positive continuous function on [a 1 ; a 2 ] such that ' is non-decreasing, is non-increasing, and let h : [a 1 ; a 2 ] ! R + be a positive continuous function. Then for a 1 < a 2 ; > 0; > 0; > 0; > 0 and s 6 = 1, we have Proof. If we multiple (12) by , then we integrate the resulting inequality with respect to v over (a 1 ; ), we obtain (13).
Remark 10. If in Theorem 9 we put = we obtain Theorem 7.
Theorem 12. Let ' be a positive decreasing continuous function on [a 1 ; a 2 ] and let h : [a 1 ; a 2 ] ! R + be a positive continuous function. Then for a 1 < a 2 ; > 0; > 0; > 0 and s 6 = 1, we have Since the function ' is positive continuous and decreasing on [a 1 ; a 2 ] then, for all 0, > 0, u; v 2 [a 1 ; ], with v u, and a 1 < a 2 , we have Multiplying both sides of (15) by F (u;s)h(u)' (u) k k ( )[F( ;u)] 1 k , then we integrate the resulting inequality with respect to u over (a 1 ; ), we get Now, multiplying both sides of (16) by 1 k , then we integrate the resulting inequality with respect to v over (a 1 ; ), it holds that (17) which implies (14).
Remark 13. This Theorem contains as a particular case Theorem 12 of [5], under the same conditions as the previous Remarks.
The latest results of our work is the extension of the previous Theorems considering the product of a family of appropriate functions.
Remark 16. If in Theorem 15 we put = we obtain Theorem 14.

CONCLUSIONS
In this work we have obtained various inequalities of the Hermite-Hadamard type, in the case of di¤erent notions of convexity, and using a generalized fractional operator, which allows obtaining as particular cases, several of those reported in the literature.
We want to point out, in addition to the observations made throughout the work, that with di¤erent choices of the F kernel we can obtain, as particular cases, several well-known integral operators. So, for example, if A) The classic Riemann integral is obtained with F (t; ) = t 1 , = 1 and = k (with notation changed). B) If F (t; ) = t 1 and = k we obtain the fractional Riemann-Liouville integral.
C) Considering F (t; ) = t s with s = 1, we can write the right sided operator as follows and similarly the left sided integral. The k-Riemann-Liouville fractional integral of Mubeen and Habibullah (see [7]). D) Katugampola fractional integral of [6] is obtained, taking F (t; ) = t (the notation is changed). E) If we put F = t s with s = 1, then we get the right sided Hadamard fractional integral of [4].
F) An integral operator with non-singular kernel can also be obtained from our De…nition 1. Thus, considering F (t; ) = exp 1 t , if = 1 we have that i , a slight modi…cation of the operator de…ned by Kirane and Toberek in [1].
From all of the above, we can conclude that many of the integral inequalities obtained in the framework of these integral operators, can be obtained as particular cases from those obtained in this work.

Authors Contribution Statement
The authors contributed equally to this work. All authors of the submitted research paper have directly participated in the planning, execution, or analysis of study.

Declaration of Competing Interest
The authors declare that there is no con ‡ict of interest regarding the publication of this article. Besides, the contents of the manuscript have not been submitted, copyrighted or published elsewhere and the visual-graphical materials such as photograph, drawing, picture, and document within the article do not have any copyright issue. Finally, all authors of the paper have read and approved the …nal version submitted.