Strongly (η,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\eta ,\omega )$\end{document}-convex functions with nonnegative modulus

We introduce a new class of functions called strongly (η,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\eta,\omega)$\end{document}-convex functions. This class of functions generalizes some recently introduced notions of convexity, namely, the η-convex functions and strongly η-convex functions. We also establish inequalities of the Hermite–Hadamard–Fejér’s type, which generalize results of Delavar and Dragomir (Math. Inequal. Appl. 20(1):203–216, 2017) and Awan et al. (Filomat 31(18):5783–5790, 2017). In addition, we obtain some new results for this class of functions. Finally, we apply our results to the k-Riemann–Liouville fractional integral operators to obtain more results in this direction.

We recapture the classical definition of convexity if the bifunction η(x, y) = xy. Recently, Delavar and Dragomir [1] obtained the following theorems for the class of η-convex functions.
where K η is an upper bound of η.
Definition 1 was further generalized by Awan et al.
The authors in [2] proved the following result.
Theorem 6 ([2]) Let F : [α, β] → R with α < β. Suppose that the function F satisfies the following conditions: (a) F is strongly η-convex with respect to modulus μ ≥ 0 and η bounded above on Then we have the following inequalities: where K η is an upper bound of η.
Stimulated by the above-mentioned work, we aim to achieve the following goals: 1. to introduce a new class of functions in Sect. 2, which generalizes preexisting notions of convexity; 2. to extend Theorems 3 and 6 to this new class of functions (see Sect. 3) and then apply the results obtained thereafter to the k-Riemann-Liouville fractional integrals; 3. finally, to establish many new integral inequalities in this direction.

A new class of convexity
We now introduce a new definition as a generalization of Definition 4.
The function F is strongly (η, ω)-convex with respect to the bifunction η(x, y) = 2x + y, ω(τ ) = τ , and modulus μ = 1. To see this, let τ ∈ [0, 1]. Then We wrap up this section by showing, by means of the next example, that the class of strongly (η, ω)-convex functions is wider than the class of strongly η-convex functions.

Main results
We break this section into three subsections. We start by presenting Hermite-Hadamard-Fejér-type results and give an application to the k-Riemann-Liouville fractional integral. Thereafter, we conclude by establishing three more theorems for the class of (η, ω)-convex functions.

Inequalities of the Hermite-Hadamard-Fejér type
Suppose that the functions F and G satisfy the following conditions: Then we have the following inequalities: where K η is an upper bound of η.
Proof For all τ ∈ [0, 1], we have Since F is strongly (η, ω)-convex, we obtain for all τ ∈ [0, 1]. Since K η is an upper bound of η, we get for all τ ∈ [0, 1]. Similarly, we can also write From this inequality we get for all τ ∈ [0, 1]. Adding (7) and (8), we obtain the following inequality for τ ∈ [0, 1]: , integrating over (0, 1) with respect to the variable τ , and using item (d) and a change of variable, we get This implies that Multiplying again (9) by G( α+β-τ (β-α) 2 ) and proceeding as before, we get Adding (10) and (11) gives which gives the first inequality. Next, we prove the second inequality. For this, let u be any element in [α, β]. Then u can be expressed as Using the strong (η, ω)-convexity of F, we obtain Multiplying this inequality by G(u) and integrating over (α, β) with respect to the variable u, we get Similarly, we can also write Applying again the strong (η, ω)-convexity of F gives Multiplying this inequality by G(u), proceeding as outlined before, and noting that where we have used the fact that Now adding (13) and (14) gives The last inequality follows by using the upper bound K η in (15): This completes the proof.
Suppose that the functions F and G satisfy the following conditions: (a) F is strongly η-convex with modulus μ ≥ 0 and η bounded above on Then we have the following inequalities: where K η is an upper bound of η.
Proof Let G : (α, β) → R be the function defined by Since H ∈ L 1 ((α, β)), it follows also that G ∈ L 1 ((α, β)). Also, by the definition of the function G we have that for u ∈ (α, β), Hence, items (c) and (d) of Theorem 10 are satisfied. Therefore, applying Theorem 10 to the function G, we get the desired inequalities.

Application to the k-Riemann-Liouville fractional operators
We start by recalling the definition of the k-Riemann-Liouville fractional integrals: the left-and right-sided k-Riemann-Liouville fractional integral operators k J α + and k J β -of order > 0 for a real-valued continuous function F(x) are defined as and where k > 0, and Γ k is the k-gamma function given by In what follows, we will need the following functions U, V, W : [α, β] → R defined by and Applying Corollary 13, we get the following result. Then we have the following inequalities: where K η is an upper bound of η.
Proof Let where , k > 0. The function H clearly satisfies the conditions of Corollary 13 since We obtain the intended inequalities by applying Corollary 13 to the function H and the following identities:

Corollary 15
Let F : [α, β] → R with α < β. Suppose that the function F satisfies the following conditions: (a) F is strongly η-convex with modulus μ ≥ 0 and η bounded above on Then we have the following inequalities: where K η is an upper bound of η.
Proof The proof follows by setting ω(τ ) = τ , τ ∈ [0, 1], in Corollary 14. For this, we notice that and thus By substituting μ = 0 in the corollary, we obtain the following result for the η-convex functions.
Suppose that the function F satisfies the following conditions: (a) F is η-convex and η bounded above on . Then we have the following inequalities: where K η is an upper bound of η.

More integral inequalities
We now proceed to obtain more results associated with this new class of functions. For this, we will need the following lemma.

Theorem 18
Assume that a function F satisfies the conditions of Lemma 17. If, in addition, |F | is strongly (η, ω)-convex on [α, β] with modulus μ ≥ 0 and ω ∈ L ∞ ([0, 1]), then for any λ ∈ R, we have the following inequalities: where h = βα, Proof We start by observing that Now using Lemma 17 and the strong (η, ω)-convexity of |F |, we get Hence the desired result is obtained by using (22) and (23).
Using again Lemma 17, the strong (η, ω)-convexity of |F | q , and the Hölder inequality, we obtain The desired result is obtained by employing identity (25).
Proof Applying Lemma 17, the strong (η, ω)-convexity of |F | q , and the Hölder inequality, we get The intended result is reached by employing identities (22) and (23).