Fractional Weighted Ostrowski-Type Inequalities and Their Applications

: An important area in the ﬁeld of applied and pure mathematics is the integral inequality. As it is known, inequalities aim to develop different mathematical methods. Nowadays, we need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the ﬁeld of inequalities due to the behavior of its deﬁnition and its properties. Furthermore, there is a strong correlation between convexity and symmetry concepts. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the last few years. In this study, by using a new identity, we establish some new fractional weighted Ostrowski-type inequalities for differentiable quasi-convex functions. Further, further results for functions with a bounded ﬁrst derivative are given. Finally, in order to illustrate the efﬁciency of our main results, some applications to special means are obtain. The obtained results generalize and reﬁne certain known results.


Introduction
Computational and Fractional Analysis are nowadays more and more in the center of mathematics and of other related sciences either by themselves because of their rapid development, which is based on very old foundations, or because they cover a great variety of applications in the real world. In current years, fractional calculus (FC) applied in many phenomena in applied sciences, fluid mechanics, physics and other biology can be described as very effective using mathematical tools of FC. The fractional derivatives have occurred in many applied sciences equations such as reaction and diffusion processes, system identification, velocity signal analysis, relaxation of damping behaviour fabrics and creeping of polymer composites [1][2][3][4]. Definition 1 ([5]). Let I be an interval of real numbers. A function ϕ : I → R is said to be convex, if for all x, y ∈ I and all t ∈ [0, 1], we have ϕ(tx + (1 − t)y) ≤ tϕ(x) + (1 − t)ϕ(y).
The concept of convex functions has been also generalized in diverse manners. One of them is the quasi-convex function defined as follows: Definition 2 ([6]). A function ϕ : I → R is said to be quasi-convex, if ϕ(tx + (1 − t)y) ≤ max{ϕ(x), ϕ(y)} holds for all x, y ∈ I and t ∈ [0, 1].
The following notations will be used in the sequel. We denote, respectively I • the interior of I and L[a, b] the set of all integrable functions on [a, b].
In [22], Alomari et al. gave the following midpoint type inequalities for differentiable quasi-convex. Theorem 2 ([22]). Let ϕ : I ⊂ [0, ∞) → R be a differentiable mapping on I • such that ϕ ∈ L[a, b], where a, b ∈ I with a < b. If |ϕ | is quasi-convex on [a, b], then the following inequality holds: , then the following inequality holds: Alomari and Darus in [23] obtained the Ostrowski-type inequalities for differentiable quasi-convex functions: , then the following inequality holds: , then the following inequality holds: Theorem 7 ([23]). Let ϕ : I ⊂ [0, ∞) → R be a differentiable mapping on I • such that ϕ ∈ L[a, b], where a, b ∈ I with a < b. If |ϕ | q is quasi-convex on [a, b], q ≥ 1, then the following inequality holds: Fractional calculus has been widely studied by many researchers over the past decades. In particular, to generalize classic inequalities. Among the best known and use of these fractional integral operators we recall that of Riemann-Liouville.
In this paper, we establish a new identity and then apply it to derive new weighted Ostrowski-type inequalities for quasi-convex functions. Further results for functions with a bounded first derivative will be given. In order to illustrate the efficiency of our main results, some applications to special means will be obtain.

Main Results
For brevity, we will used in the sequel J = [a, b]. In order to find our main results, we need to prove the following lemma. and Now, multiplying (4) by Similar work give Multiplying (6) by By taking the difference between (5) and (7), we get which is the desired result.
Proof. From Lemma 1, properties of modulus and quasi-convexity of |ϕ |, we have The proof is completed.

Corollary 2.
Choosing ω(u) = 1 b−a in Theorem 8, we get Moreover, if we take x = a+b 2 , we obtain Proof. Applying properties of modulus, Lemma 1, Hölder's inequality, and quasi-convexity of |ϕ | q , we obtain The proof is completed.
Proof. By properties of modulus, applying Lemma 1, power mean inequality and quasiconvexity of |ϕ | q , we obtain The proof is completed.

Corollary 9.
Choosing ω(u) = 1 b−a in Theorem 10, we have Moreover, if we take x = a+b 2 , we obtain

Further Results
In this section, we will prove the following results.
Proof. Applying Lemma 1, we get From (9), we have where Λ α (a, x, b, ω, ϕ) is defined as in (8). Applying absolute value on both sides of (10) and using the fact that m ≤ ϕ (x) ≤ M for all x ∈ J, we obtain The proof is completed.
Corollary 11. For x = a+b 2 in Theorem 11, we have

Corollary 12.
Putting ω(u) = 1 b−a in Theorem 11, we get Moreover, if we take x = a+b 2 , we obtain
Proof. Applying Lemma 1, we have From (12), we get where Ξ α (a, x, b, ω, ϕ) is defined as in (11). Applying absolute value on both sides of (13) and r-H-Hölder property of ϕ , we obtain The proof is completed.
Corollary 15. Letting x = a+b 2 in Theorem 12, we have

Corollary 16.
Choosing ω(u) = 1 b−a in Theorem 12, we get Moreover, if we take x = a+b 2 , we obtain .

Corollary 18.
In Corollary 17, if we choose ω(u) = 1 b−a , we obtain Moreover, if we take x = a+b 2 , we have 16 .
Corollary 20. For x = a+b 2 in Corollary 19, we get Let recall from [23] the following quasi-convex functions, ϕ(u) = u n , n ∈ N, n ≥ 2 and ϕ(u) = 1 u for all u > 0, respectively. Using Section 2, we are in position to prove the following results regarding above special means. Proposition 1. Let a, b ∈ R, 0 < a < b, n ∈ N and n ≥ 2, then Proof. Taking ϕ(u) = u n , u > 0 for x = a+b 2 and ω(u) = u in Corollary 3, we get the desired result. Proposition 2. Let a, b ∈ R, 0 < a < b, n ∈ N and n ≥ 2, then .

Conclusions
The main results and future research of the article can be summarized as follows: • A new identity regarding fractional weighted Ostrowski-type is established. • New fractional weighted Ostrowski-type inequalities for quasi-convex functions using the above identity are deduced. • Several further results for function with a bounded first derivative are given. • Some applications to special means are obtained. • The efficiency of our results is shown. • As future research, from our results, interested reader can find several new interesting inequalities from many areas of pure and applied sciences. Moreover, they can derive (using our technique) applications to special means for different quasi-convex functions.