k-fractional integral trapezium-like inequalities through (h,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(h,m)$\end{document}-convex and (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 mappings

In this paper, a new general identity for differentiable mappings via k-fractional integrals is derived. By using the concept of (h,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(h,m)$\end{document}-convexity, (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}-convexity and the obtained equation, some new trapezium-like integral inequalities are established. The results presented provide extensions of those given in earlier works.


Introduction
Let f : I ⊆ R → R be a convex mapping and a, b ∈ I along with a < b. The inequality named Hermite-Hadamard's inequality, is one of the most famous results for convex mappings. This inequality (1.1) is also known as trapezium inequality.
The trapezium-type inequality has remained an area of great interest due to its wide applications in the field of mathematical analysis. Many researchers generalized and extended it via mappings of different classes. For recent results, for example, see [1][2][3][4][5][6][7] and the references mentioned in these papers.
In 2013, Sarikaya et al. [8] established the following theorem by utilizing Riemann-Liouville fractional integrals.  [a, b]. Suppose that f is a convex function on [a, b], then the following inequalities for fractional integrals hold:

2)
where the symbols J μ a + f and J μ b -f denote respectively the left-sided and right-sided Riemann-Liouville fractional integrals of order μ > 0 defined by Here, (μ) is the gamma function and its definition is (μ) = ∞ 0 e -t t μ-1 dt. It is to be noted In the case of μ = 1, the fractional integral recaptures the classical integral. Because of the extensive application of Riemann-Liouville fractional integrals, some authors extended their studies to fractional trapezium-type inequalities via mappings of different classes. For example, refer to [9][10][11][12] for convex mappings, to [13] for s-convex mappings, to [14] for (s, m)-convex mappings, to [15] for r-convex mappings, to [16] for harmonically convex mappings, to [17] for s-Godunova-Levin mappings, to [18,19] for preinvex mappings, to [20] for MT m -preinvex mappings, to [21] for h-convex mappings and to references cited therein.
In [22], Mubeen and Habibullah introduced the following class of fractional derivatives.
The concept of k-Riemann-Liouville fractional integral is an important extension of Riemann-Liouville fractional integrals. We want to stress here that for k = 1 the properties of k-Riemann-Liouville fractional integrals are quite dissimilar from those of general Riemann-Liouville fractional integrals. For this, the k-Riemann-Liouville fractional integrals have aroused the interest of many researchers. Properties concerning this operator can be sought out [23][24][25][26], and for the bounds for integral inequality related to this operator, the reader can refer to [27][28][29] and the references mentioned in these papers.
Motivated and inspired by the recent research in this field, we obtain some k-Riemann-Liouville fractional integral of trapezium-type inequalities for (h, m)-convex mappings and (α, m)-convex mappings. The results presented in this paper provide extensions of those given in earlier works.
To end this section, we restate some special functions and definitions.
Note that, if we choose m = 1 in Definition 1.7, f reduces to a tgs-convex function in Definition 1.5.

A lemma
To prove our main results, we consider the following new lemma. Lemma 2.1 Let f : I ⊆ R → R be a differentiable mapping on I o (the interior of I) with 0 ≤ a < mr, a, r ∈ I, for some fixed m ∈ (0, 1]. If f ∈ L 1 [a, mr], then the following equality for k-fractional integral along with λ ∈ (0, 1]\ 1 2 , k > 0 and μ > 0 exists: where T k,μ (m, λ, r) Proof It suffices to note that Integrating by parts, we get , equality (2.4) can be written as and similarly we get Similarly, taking λ = 1 in Lemma 2.1, we obtain , it is easy to see that identity (2.8) is equal to identity (2.7).

Remark 2.1
(i) In Corollary 2.1, if we put r = b, then one can obtain Lemma 3.1 which is proved in [35]. Further, if we take m = 1, then we obtain Lemma 2.1 in [12].

k-fractional integral inequalities for (h, m)-convex functions
In what follows, we establish some k-fractional integral inequalities for (h, m)-convex functions by using Lemma 2.1.
Now, we point out some special cases of Theorem 3.1.
Especially if we put k = 1, we obtain Theorem 3.2 in [35].

Corollary 3.4
In Theorem 3.1, if we put h(t) = 1, then we obtain the following inequality for (m, P)-convex functions: Especially if we choose m = 1 and λ = 1 or λ = 0, we have

Corollary 3.6 In Theorem 3.1, if we choose h(t) = t(1t), then we obtain the following inequality for (m, tgs)-convex functions:
Especially if we put m = 1 and λ = 1 or λ = 0, we get t , then we obtain the following inequality for m-MT-convex functions: Especially if we put m = 1 and λ = 1 or λ = 0, we get . Now, we prepare to introduce the second theorem as follows.
Proof Using Lemma 2.1, Hölder's inequality and the (h, m)-convexity of |f | q , we have T k,μ (m, λ, r) Here, we use (A -B) q ≤ A q -B q for any A ≥ B ≥ 0 and q ≥ 1.
Let us point out some special cases of Theorem 3.2.
Especially if we put m = 1 and λ = 0 or λ = 1, we have

Corollary 3.10 In Theorem 3.2, if we put h(t) = t(1t), then we get the following inequality for (m, tgs)-convex functions:
Especially if we put m = 1 and λ = 0 or λ = 1, we have t , then we get the following inequality for m-MT-convex functions: Especially if we put m = 1 and λ = 0 or λ = 1, we have . Now, we are ready to state the third theorem in this section.
Proof Applying Lemma 2.1, Hölder's inequality and the (h, m)-convexity of |f | q , we have T k,μ (m, λ, r) Here, we use the fact that (A -B) q ≤ A q -B q for any A ≥ B ≥ 0 and q ≥ 1, which completes the proof.
Now, we point out some special cases of Theorem 3.3.

Corollary 3.12
In Theorem 3.3, if we choose h(t) = t and r = b, then we obtain the following inequality for m-convex functions: Especially if we put k = 1, we obtain Theorem 3.3 in [35]. Further, if we put m = 1, we obtain Theorem 2.6 in [12].

Remark 3.2 In Corollary 3.13,
(a) if we take k = 1 and r = b, we can get Corollary 2.7 in [12], (b) if we take k = 1 = μ and r = b, we can get Corollary 2.8 in [12].
Especially if we put m = 1 and λ = 0 or λ = 1, then we have

k-fractional inequalities for (α, m)-convex functions
Using Lemma 2.1 again, we state the following theorems.