Some fractional Hermite–Hadamard-type integral inequalities with s-(α,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha,m)$\end{document}-convex functions and their applications

Under the new concept of s-(α,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha,m)$\end{document}-convex functions, we obtain some new Hermite–Hadamard inequalities with an s-(α,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha,m)$\end{document}-convex function. We use these inequalities to estimate Riemann–Liouville fractional integrals with second-order differentiable convex functions to enrich the Hermite–Hadamard-type inequalities. We give some applications to special means.


Introduction
Convex functions are a kind of important functions widely used in mathematical programming. They are not only closely related to continuity and differentiability, but also play important roles in inequalities. Therefore convex functions has been widely used in many research fields such as life and management science, optimization [1,2], and so on. In optimization inequalities, generalized classical convexity is often used together with convexity theory and inequality theory, in which Hermite-Hadamard integral inequalities containing convex functions are valued by many mathematicians because of their pertinence and ease of use. The classical Hermite-Hadamard-type integral inequality is the following [3]: Let g : I ⊆ R → R be a convex function on the interval I of real numbers, and let c, d ∈ I with c < d. Then function on I 0 , and let a, b ∈ I 0 with a < b. If |g | is convex on [a, b], then we have the following inequality: Let g : I 0 ⊆ R → R be a differentiable function on I 0 , let a, b ∈ I 0 with a < b, and let p > 1. If the function |g | p/p-1 is convex on [a, b], then we have the following inequality: Khaled and Agarwal [6] extended the interval [a, b] and made new estimates of the Hermite-Hadamard inequality on the interval [ 3a-b 2 , 3b-a 2 ]: Let g : I ⊆ R → R be a differentiable function on I, let a, b ∈ I with a < b, and let its derivative g : [ 3a-b 2 , 3b-a 2 ] → R be a continuous function on [ 3a-b 2 , 3b-a 2 ]. Let q ≥ 1. If |g | q is a convex function on [ 3a-b 2 , 3b-a 2 ], then we have the following inequality: Özcan and Íscan [7] generalized the Hermite-Hadamard inequality for s-convex functions. Let g : I ⊆ R → R be a differentiable function on I, and let a, b ∈ I with a < b. If g ∈ L[a, b], then we have the following inequality: All these different estimates of integral inequalities of integer order hold under the convexity of |g |.
With the in-depth study of integer-order Hermite-Hadamard inequality, more and more scholars have also done a lot of research and extensions of fractional Hermite-Hadamard integral inequality, among which there are many papers related to the Riemann-Liouville fractional integral. Sarikaya et al. [8] studied the Hermite-Hadamard integral inequality to estimate arithmetic means and Riemann-Liouville fractional integrals using a convex function |g |: If g is a convex function on [a, b], then we have the following inequalities for fractional integrals: Let g : , then we have the following inequalities for fractional integrals: Chun et al. [9] studied the Hermite-Hadamard integral inequality to estimate geometric means and Riemann-Liouville fractional integrals using a convex function |g |: , then we have the following inequalities for fractional integrals: Li Xiaoling and Shahid [10] studied the Hermite-Hadamard inequality of s-(α, m)convex functions with parameter Riemann-Liouville fractional integral: Let g : [c, d] → R be a differentiable function on [c, d] with c < d such that g is s-(α, m)convex on [a, b]. Then we have the following inequality for Riemann-Liouville fractional integrals with 0 < α ≤ 1: There are many other Hermite-Hadamard integral inequalities for convex functions; we refer the interested readers to [11][12][13][14][15][16][17][18][19][20][21][22].
In [10] the author studies the inequalities of first-order differentiable convex functions on the right side of the Hermite-Hadamard inequality. In this paper, using s-(α, m)-convex functions and Riemann-Liouville fractional integrals, we study some Hermite-Hadamard inequalities of second-order differentiable convex functions on the right side of the inequality and apply them to special means.
The arrangement of this paper is as follows. In Sect. 2, we introduce the classes of convex functions to prepare the work; In Sect. 3, we prove new Hermite-Hadamard integral inequalities using new concepts and the Riemann-Liouville fractional integral; In Sect. 4, we apply the results to special mean values.

Preliminaries
In this section, we recall some important definitions and results.
The general classical convexity is defined as follows.
The left-sided and right-sided Riemann-Liouville fractional integrals of order α > 0, with a ≥ 0, are defined by and where (α) = ∞ 0 e -u u α-1 du. In the case α = 1, the fractional integral reduces to the classical integral. Properties relating to this operator can be found in [25].

Main result and proof
In [10], all the Hermite-Hadamard integral inequalities were based on the s-(α, m)convexity of |g |. If we do not know the convexity of |g |, but |g | is convex, then we will get new Hermite-Hadamard inequalities. Next, we will study fractional Hermite-Hadamard integral inequalities based on the convexity of |g |, where the s-(α, m)-convexity is in the first sense. First, we give a lemma, which will be used in later important conclusions. where Proof The proof is obtained by integration by parts based on equation (16). We have Let u = tc + (1t) c+d 2 . Then Using the same algorithm, we get: Multiplying both sides by (d-c) 2 2 α+2 , we get (17). This completes the proof. where , then for all t ∈ [0, 1], by Lemma 3.1 we obtain: , .
Summing the four terms on the right-hand side of the inequality, we get (18). This completes the proof.
Let α = s = m = 1 in Theorem 3.1. Then (18) reduces to an integer-order inequality of general convexity.

Corollary 3.1 Let g, g be defined as in Theorem
In [8] the author used the convexity of |g | to estimate the error. We can do a similar work by using the convexity of |g |.
Taking α = s = m = 1 in Theorem 3.2, we get the following integer-order inequalities of general convexity. First, taking λ = 0, we get the following: Second, taking λ = 1, we get the following: where Proof Using the concavity of |g | q and the power-mean inequality, we obtain Then so that |g | is also concave. By the Jensen integral inequality for concave functions This completes the proof.
Taking α = 1 in Theorem 3.3, we get the following integer-order inequalities. First, taking λ = 0, we get the following:  2

Applications of the result
Using the results obtained, we can get new estimates for the following special means.