Article Simpson Type Conformable Fractional Inequalities

In this study, a new Simpson type conformable fractional integral equality for convex functions is established. Based on this identity, some results related to Simpson-like type inequalities are obtained. Also, some estimation results are given for the special cases of the derivative of a function used in our results, and some applications are presented for special means such as the arithmetic, geometric, and logarithmic means.


Introduction
Inequalities are extremely useful in mathematics, especially when we deal with quantities that we do not know exactly what they equate too. Often, one can solve a mathematical problem, by estimating an answer, rather than writing down exactly what it is. For more information in this regard, one can see [1][2][3][4][5][6][7][8].
Fractional calculus has been a fascinating area for many researchers in the past and present eras. In the recent two decades, the use of fractional calculus in both pure and applied disciplines of science and engineering has increased significantly. The inequalities involving fractional integrals have become a noticeable approach in recent decades and have acted as a powerful tool for numerous investigations. In recent years, various types of new fractional integral inequalities including Hermite-Hadamard type inequalities have been established via convexity, which provides quite helpful and valid applications in areas such as probability theory, functional inequalities, interpolation spaces, Sobolev spaces, and information theory (see the papers [9,10]).
The concept of convexity is not a new one even it occurs in some other form in Archimedes' treatment of orbit length. Convex geometry is now a mathematical field in its own right, and significant results have been made in various modern studies such as real analysis, functional analysis, and linear algebra by employing the concept of convexity (see [11][12][13][14]). In the last few decades, the subject of convex analysis has got rapid development in view of its geometry and its role in the optimization.
The modern theory of inequalities is another attractive area for researchers in which the notion of convexity plays a major role in improving the estimation bounds of various types of integral inequalities. The following inequality which is known as Simpson's inequality has been studied by several authors (see the papers [15][16][17][18][19][20]). Theorem 1. Let ψ : ½γ, δ ⟶ ℝ be a four times continuously differentiable mapping on ðγ, δÞ and kψ ð4Þ k ∞ = sup jψ ð4Þ ðεÞj < ∞: Then, The definition below is given in [10,11].
We obtain a new Simpson type identity in this study and use it to derive some results about Simpson-like type inequalities through using conformable fractional integral with some applications.

Preliminaries
In this section, we give some definitions and basic results which are useful in obtaining the main results.
Moreover, the papers in [3,9,13] contain additional information on conformable fractional integrals. The following are the definitions of beta and incomplete beta functions, as well as the relationship between the gamma and beta functions, as stated in [21].

Main Results
To obtain the main results, first, we need to prove the following lemma: Lemma 5. Let ψ : I ⊂ ð0,∞Þ ⟶ ℝ be a differentiable function on I ∘ , γ, δ ∈ I ∘ and γ < δ: If ψ′∈L½γ, δ, then Proof. We start by considering the following computations which follows from change of variables and using the definition of the conformable fractional integrals.
Remark 8. If we take τ = m + 1, and after that if we take τ = 1 in Theorem 7, we obtain the inequality Corollary 1 in [15].
Remark 10. If we take τ = m + 1, and after that if we take τ = 1 in Theorem 9, we obtain the inequality Corollary 4 in [15].

Journal of Function Spaces
Proof. From Lemma 5 and using the power mean inequality, we have that the following inequality holds: By the convexity of jψ ′ j q , Using the last two inequalities, we obtain the inequality (15).
Remark 14. If we take τ = m + 1, and after that if we take τ = 1 in Theorem 13, we obtain the inequality Corollary 4 in [15].

Estimation Results
If the function ψ ′ is bounded, then we have the next result.