Some New Ostrowski-Type Inequalities Involving σ-Fractional Integrals

Inequalities play a pivotal role in modern analysis. Mathematical analysis depends upon many inequalities. In recent years, an extensive research has been carried out on obtaining various analogues of classical inequalities using different approaches, for details and applications, see [1–4]. A very interesting approach is to obtain fractional analogues of the inequalities. +e fractional version of inequalities plays a significant role in the establishment of the uniqueness of solutions for certain fractional partial differential equations. Sarikaya et al. [5] were the first to introduce the concepts of fractional calculus in the theory of integral inequalities by obtaining the fractional analogues of classical Hermite–Hadamard’s inequality. Dragomir [6, 7] obtained fractional versions of Ostrowski-like inequalities. Erden et al. [8] recently obtained some more new fractional analogues of Ostrowski-type inequalities using bounded functions. Sarikaya [9] introduced the notion of twodimensional Riemann–Liouville fractional integrals and obtained some new fractional variants of Hermite–Hadamard’s inequality on two dimensions. Having inspiration from the research work of Mubeen and Habibullah [10] and Sarikaya [9], Awan et al. [11] introduced the concepts of σ-Riemann–Liouville fractional integrals on two dimensions and obtained twodimensional fractional integral inequalities. It is worth to mention here that if σ⟶ 1, then σ-Riemann–Liouville fractional integrals reduces to classical Riemann–Liouville fractional integral. Note that the concept of σ-Riemann–Liouville fractional integral is a significant generalization of classical Riemann–Liouville fractional integrals; as for σ ≠ 1, the properties of σ–Riemann–Liouville fractional integrals are quite different from the classical Riemann–Liouville fractional integrals. +e aim of this paper is to obtain some new fractional analogues the classical Ostrowski’s inequality using the concepts of σ-fractional integrals. In order to obtain the main results of the paper, we first derive some new lemmas results, and then using these lemmas as auxiliary results, we derive our main results of the paper. Let us first recall some previously known concepts and results. +e first one is the definition of the Riemann–Liouville fractional integrals.


Introduction and Preliminaries
Inequalities play a pivotal role in modern analysis. Mathematical analysis depends upon many inequalities. In recent years, an extensive research has been carried out on obtaining various analogues of classical inequalities using different approaches, for details and applications, see [1][2][3][4]. A very interesting approach is to obtain fractional analogues of the inequalities. e fractional version of inequalities plays a significant role in the establishment of the uniqueness of solutions for certain fractional partial differential equations. Sarikaya et al. [5] were the first to introduce the concepts of fractional calculus in the theory of integral inequalities by obtaining the fractional analogues of classical Hermite-Hadamard's inequality. Dragomir [6,7] obtained fractional versions of Ostrowski-like inequalities. Erden et al. [8] recently obtained some more new fractional analogues of Ostrowski-type inequalities using bounded functions. Sarikaya [9] introduced the notion of twodimensional Riemann-Liouville fractional integrals and obtained some new fractional variants of Hermite-Hadamard's inequality on two dimensions. Having inspiration from the research work of Mubeen and Habibullah [10] and Sarikaya [9], Awan et al. [11] introduced the concepts of σ-Riemann-Liouville fractional integrals on two dimensions and obtained twodimensional fractional integral inequalities. It is worth to mention here that if σ ⟶ 1, then σ-Riemann-Liouville fractional integrals reduces to classical Riemann-Liouville fractional integral. Note that the concept of σ-Riemann-Liouville fractional integral is a significant generalization of classical Riemann-Liouville fractional integrals; as for σ ≠ 1, the properties of σ-Riemann-Liouville fractional integrals are quite different from the classical Riemann-Liouville fractional integrals. e aim of this paper is to obtain some new fractional analogues the classical Ostrowski's inequality using the concepts of σ-fractional integrals. In order to obtain the main results of the paper, we first derive some new lemmas results, and then using these lemmas as auxiliary results, we derive our main results of the paper.
Let us first recall some previously known concepts and results.
e first one is the definition of the Riemann-Liouville fractional integrals.
en, the Riemann-Liouville integrals of order α 1 > o with a > 0 are defined as follows: respectively. Here, Γ(α 1 ) is the gamma function. ese integrals are motivated by the well-known Cauchy formula: Mubeen and Habibullah [10] introduced the σ-Riemann-Liouville fractional integrals as follows.
e σ-Reimann-Liouville fractional integrals σ J α 1 a + Ξ and σ J α 1 b − Ξ of order α 1 > 0 with a ≥ 0 and σ > 0 are defined as follows: e above integrals for all functions are continuous and integrable on the interval (0, ∞). Note that if f ∈ L 1 [a, b] and a > 0, then σ J α 1 a exists almost everywhere on [a, b]. If For more details, see [13].
For the sake of simplicity, we define the following functions as

Key Lemmas.
In this section, we prove some lemmas which will help us in obtaining the main results of the paper.

Lemma 1.
Let Ξ: D ⟶ R be an absolutely continuous, differentiable function such that (z 2 Ξ(θ, μ)/zθzμ) exists and is continuous on D⊆R 2 . en, for any (x, y) ∈ D, we have where Proof. Now, is implies Journal of Mathematics Now, consider Similarly, Using (10)- (13) in (9), we get the required result. □ Lemma 2. Let Ξ: D ⟶ R be an absolutely continuous, differentiable function such that (z 2 Ξ(θ, μ)/zθzμ) exists and is continuous on D⊆R 2 . en, for any (x, y) ∈ D, we have y; a, b, c, d), where Proof. e proof is same as the proof of Lemma 1.

Journal of Mathematics
Now, Similarly, Using the values of I 1 , I 2 , I 3 , and I 4 in (17), we get the required result. □ 2.2. Results and Discussion. In this section, we discuss our main results.