New generalization of reverse Minkowski's inequality for fractional integral

The realizations of inequalities which containing the fractional integral and di erential operators is considered to be important due to its wide implementations among authors. In this research, we introduce some new fractional integral inequalities of Minkowski's type by using Riemann-Liouville fractional integral operator. We replace the constants appears on Minkowski's inequality by two positive functions. Further, we establish some new fractional inequalities related to the reverse Minkowski type inequalities via RiemannLiouville fractional integral. Using this fractional integral operator, some special cases of reverse Minkowski type are also discussed. Mathematics Subject Classi cation (2010): 26D10, 26A33


Introduction
During their unrelenting eort in the development of mathematics, mathematicians in the past few decades have expanded the concept of classical calculus of derivatives and integrals for integer orders to the fractional calculus, which is a generalized form of classical integrals and derivatives in case of non-integer order. Recently, the fractional calculus theory has get more attention due to its important applications in various elds such as computer networking, biology, physics, uid dynamics, signal processing, image processing, control theory and other elds. One of the widespread approaches among authors is the use of fractional integrals and derivatives operators. As a consequence many dierent kinds of fractional integrals and derivatives have been realized, such as the Riemann-Liouville, Weyl types, Liouville, Hadamard, Katugampola and some other types can be found found in Kilbas et.al. [13].
In classical integral and dierential equations, mathematical inequalities play a responsible role and in the last few years, a number of useful and important mathematical inequalities invented by many authors. Inequalities which involve fractional integrals and derivatives are crucial in the study of a number of differential and integral equations, among which we mention [1], [2], [3], [5], [17]. One of the most renowned and important integral inequality is given by Hermann Minkowski. In the last few decades, this inequality has received considerable attention from many authors and several articles have appeared in the literature. In (2010), Dahmani [9], gave the reverse Minkowski and Hermite-Hadamard inequalities by mean of RiemannLiouville fractional integral. In (2010) Our purpose in this paper is to use Riemann-Liouville fractional integral operator to introduce some new fractional integral inequalities of Minkowski's type in case of functional bounds. Moreover, we establish some new fractional inequalities related to the reverse Minkowski's type inequality via Riemann-Liouville fractional integral operator. The paper is organized as follows: In second section, we recollect some notations, denitions, results and preliminary facts which are used throughout this paper. In third section, we give our main results of reverse Minkowski's inequality with functional bounds. In fourth section, we present some other related results involving Riemann-Liouville fractional integral operator.

Basic Denitions and Tools
Now, in this section, we give some basic denitions and properties of fractional integrals used to obtain and discuss our new results.
[13] Let δ > 0 and f be an integrable functions on [a, b] with a ≥ 0. The notation I δ a + and I δ b − are called respectively the left and right-sided Riemann-Liouville fractional integrals and dened by and where, Γ (δ) = ∞ 0 e −u u δ−1 du is a Gamma function and I 0 . In present paper, we use only the left-sided fractional integrals (1) to obtain and discuss our results. For the convenience of establishing the results, we use the expression I δ to denote the left-sided Riemann-Liouville fractional integral operator I δ a + at a = 0.
[6] Let n ≥ 1. Assume that there exist two positive functions f, g dened on For the inequalities (3) and (4), Dahmani in [9] established a fractional versions inequalities as follows Theorem 2.5. Let δ > 0, n ≥ 1. Assume that there exist two positive Note that, the corresponding results of all results discussed in this paper for the right-sided Riemann-Liouville fractional integrals (2) can be obtained by same arguments.

Reverse Minkowski's inequalities for fractional integral
Here, we are ready to give our generalization of the reverse Minkowski inequalities for fractional integrals in case of functional bounds.
Then for all δ > 0, n ≥ 1, we have Proof. Using the condition f (τ ) and by using the condition Multiplying both sides of (8) and both sides of (9) by 1 Γ(δ) (x − τ ) δ−1 , τ ∈ (0, x) and integrating the resulting inequalities with respect to τ over (0, x) , we get respectively and which can be written as and Now, multiplying by 1 (12) and (13), then integrating the resulting inequalities with respect to ρ over (0, x) , we obtain respectively which yields and Hence, the required inequality (7) can be obtained by adding the inequalities (14) and (15), which completes the proof.
In the following corollary, we apply Theorem (3.1) for two parameters Corollary 3.2. Let f, g be two positive functions Then for all δ > 0, γ > 0, n ≥ 1, we have Proof. The proof follows by multiplying both sides of (12) and (13) by 1 Γ(γ) (x − ρ) γ−1 , ρ ∈ (0, x) and integrating the resulting inequalities with respect to ρ over (0, x) , then the proof can be completed with same argument as in Theorem3.1.
Theorem 4.2. Let f, g be two positive functions on [0, ∞) , such that I δ f n (x) , I δ g n (x) < ∞, ∀x ∈ [0, ∞) . Assume that there exist two positive functions u, v such that 0 Then for all δ > 0, n ≥ 1, we have Proof. Using the condition 0 < k < u (ρ) ≤ f (τ ) Also, we have Multiplying both sides of each of (37) and (38) by 1 Γ(δ) (x − τ ) δ−1 , τ ∈ (0, x) and integrating the resulting inequalities with respect to τ over (0, x) , we obtain respectively it follows that Conclusion 4.4. In this paper, we have presented the reverse Minkowski's inequalities in new generalization through replacing the constants which appear as borders on Minkowski's inequality by two positive functions. Our work produces functional bounds analogues of many pre-existing results in the literature. We have also presented some other related inequalities for reverse Minkowski type inequalities. The denitions and a few advantages of the used fractional integral over the other fractional integrals are presented in the literature .