Hermite–Jensen–Mercer type inequalities for conformable integrals and related results

In this paper, certain Hermite–Jensen–Mercer type inequalities are proved via conformable integrals of arbitrary order. We establish some different and new fractional Hermite–Hadamard–Mercer type inequalities for a differentiable function f whose derivatives in the absolute values are convex.


Introduction
The concept of convex function differs from other function classes with its features such as high application areas in mathematics, statistics, and many other applied sciences. This is due to its special useful definition having geometric interpretation. Moreover, it is one of the indispensable parts of inequality theory and has become the main motivation point of many inequalities.
Although the concept of convex function has a useful place in many fields of mathematical analysis and statistics, it has revealed its main importance and effectiveness in the field of inequality theory with convex analysis. Many classical and analytical inequalities, especially Hadamard's inequality, Jensen's inequality, and Steffensen's inequality, have been achieved with the help of this concept. Detailed information and effectiveness of this function class can be found in [1][2][3][4][5][6].
Based on this useful inequality, several papers have been performed. One of them can be stated in Matkovic et al. This study includes some new findings on Jensen's inequality of Mercer type for operators with applications [9]. Then, in 2009, Mercer's result was expanded to higher dimensions by Niezgoda's paper in [10]. Recently, notable contributions have been made on Jensen's Mercer type inequality. In 2014, Kian gave a concept of Jensen's inequality for superquadratic functions [11]. Therefore, Anjidani proved some motivating results on reverse Jensen-Mercer type operator inequalities and Jensen-Mercer operator inequalities for superquadratic functions (see [12,13]). Ali and Khan generalized integral Mercer's inequality and integral means in [14].
Remark 1.1 Notice that the conformable derivatives of order θ > 1 have memory effect with kernel whose power law is integer.
In this article, by using the Jensen-Mercer inequality, we prove Hermite-Hadamard type inequalities for fractional integrals, and we establish some new conformable integrals connected with the left and right sides of Hermite-Hadamard type inequalities for differentiable mappings whose derivatives in absolute value are convex. Moreover, there will be further equalities for differentiable functions using Hölder inequality and power mean inequality.

Hermite-Hadamard-Mercer type inequalities for conformable integrals
By using Jensen-Mercer inequality, Hermite-Hadamard type inequalities can be expressed via conformable integrals as follows.
Remark 2.1 For α = n + 1 in Theorem 2.1, we get Theorem 3 proved in [28] in the integer case order.
Proof To prove the first part of the inequality, by using the Jensen-Mercer's inequality and by changing the variables τ = κ 2 μ + 2-κ 2 ν and ω = 2-κ 2 μ + κ 2 ν, κ ∈ [0, 1], we can write the following inequality for ∀τ , ω ∈ [θ , ϑ]: Multiplying both sides by 1 n! κ n (1κ) α-n-1 and then integrating the resulting inequality and so the first inequality of (2.7) is proved. Now, for the proof of the second inequality of the theorem, we first note that if φ is a convex function, then for κ ∈ [0, 1] it gives By adding the inequalities of (2.8) and (2.9), we have Multiplying both sides by 1 n! κ n (1κ) α-n-1 and then integrating the resulting inequality over κ ∈ [0, 1], we have Multiplying the above inequality by 1 2 , we obtain After further simplifications, we get the required inequality.
Proof It suffices to note that where and By combining (2.12) and (2.13) with (2.11), we get (2.10).
Remark 2.2 If we set μ = a and ν = b in Lemma 2.1, we will get Lemma 3.1 in [29].
,then the following inequality for conformable integrals holds: 1], and B(·, ·) is the Euler beta function.
Proof By using Lemma 2.1 and Jensen-Mercer's inequality, we have where , On the other hand, using the property of incomplete beta function, we have which completes the proof.
Proof Integrating by parts and changing the variables with u = θ + ϑ -( κ 2 μ + 2-κ 2 ν), we get the following results via conformable integrals: Similarly, Proof Taking modulus in Lemma 2.2 and using the well-known power mean inequality with convexity of |φ | q and Jensen-Mercer's inequality, we have (2.20) Integrating by parts, we get the following equalities: Proof Taking modulus in Lemma 2.2 and using the well-known Hölder inequality with convexity of |φ | q and Jensen-Mercer's inequality, we have After some basic calculations, we get the required result.
Proof Integrating by parts and changing the variables with u = θ + ϑ -( κ 2 μ + 2-κ 2 ν), we get the following results via conformable integrals: (2.26) Similarly, Adding equations (2.26), (2.27) and multiplying with (ν-μ) 2 8 , we get the desired result.  Proof Taking modulus in Lemma 2.3 and using the well-known power mean inequality with convexity of |φ | q and Jensen-Mercer's inequality, we have (2.29) By using Lemma 2.2 in [31], we get the following equalities: Proof Taking modulus in Lemma 2.3 and using the well-known Hölder inequality with convexity of |φ | q and Jensen-Mercer's inequality, we have After computing the above integrals, we get the required result.
Remark 2.10 If we set μ = θ and ν = ϑ in Theorem 2.8, we get Theorem 2.2 for the case of m = 1 in [31].
Proof By using Lemma 2.1 with Jensen-Mercer's inequality, the convexity of |φ | q and applying the Hölder-İşcan integral inequality that is given in (Theorem 2.1, [32]), we can write B(n + 1, αn) By making use of the computations, one can have the required result.
If |φ | q is a convex function on [θ , ϑ], then the following inequality for conformable integrals holds:   [33], so that certain sequential conformable integrals become special cases of them. However, higher order conformable integrals, for which we have proved Hermite-Jensen-Mercer type inequalities in this work, have a different structure and cannot be considered as special cases of the nonlocal fractional ones. This observation, besides the fact that the conformable integrals with order larger than 1 have kernels of integer power law, adds more interest to the proven results in this article. In fact, this inequality work, to the best of our knowledge, is one among few for such higher order extension.