Hermite-Hadamard-type inequalities for (g,φh)-convex dominated functions

In this paper, we introduce the notion of (g,φh)-convex dominated function and present some properties of them. Finally, we present a version of Hermite-Hadamard-type inequalities for (g,φh)-convex dominated functions. Our results generalize the Hermite-Hadamard-type inequalities in Dragomir et al. (Tamsui Oxford Univ. J. Math. Sci. 18(2):161-173, 2002), Kavurmacı et al. (New Definitions and Theorems via Different Kinds of Convex Dominated Functions, 2012) and Özdemir et al. (Two new different kinds of convex dominated functions and inequalities via Hermite-Hadamard type, 2012). MSC:26D15, 26D10, 05C38.


Introduction
The inequality which holds for all convex functions f : [a, b] → R, is known in the literature as Hermite-Hadamard's inequality.
In [1], Dragomir and Ionescu introduced the following class of functions. for all x, y ∈ I and λ ∈ [0, 1] .
In [2], Dragomir et al. proved the following theorem for g−convex dominated functions related to (1.1).
Let g : I → R be a convex fuction and f : I → R be a g−convex dominated mapping. Then, for all a, b ∈ I with a < b, In [1] and [2], the authors connect together some disparate threads through a Hermite-Hadamard motif. The first of these threads is the unifying concept of a g−convex-dominated function. In [3], Hwang et al. established some inequalities of Fejér type for g−convex-dominated functions. Finally, in [4], [5] and [6] authors introduced several new different kinds of convex -dominated functions and then gave Hermite-Hadamard-type inequalities for this classes of functions.
In [7], S. Varošanec introduced the following class of functions. I and J are intervals in R, (0, 1) ⊆ J and functions h and f are real non-negative functions defined on J and I, respectively. Definition 2. Let h : J → R be a non-negative function, h ≡ 0. We say that f : I → R is an h−convex function, or that f belongs to the class SX (h, I) , if f is non-negative and for all x, y ∈ I, α ∈ (0, 1], we have If the inequality (1.2) is reversed, then f is said to be h−concave, i.e. f ∈ SV (h, I) .
Youness have defined the ϕ−convex functions in [9]. A function ϕ : and t ∈ [0, 1] the following inequality holds: In [8], Sarıkaya defined a new kind of ϕ−convexity using h−convexity as following: Definition 4. Let I be an interval in R and h : (0, 1) → (0, ∞) be a given function. We say that a function f : for all x, y ∈ I and t ∈ (0, 1) .
In the following sections our main results are given: We introduce the notion of (g, ϕ h ) −convex dominated function and present some properties of them. Finally, we present a version of Hermite-Hadamard-type inequalities for (g, ϕ h ) −convex dominated functions. Our results generalize the Hermite-Hadamard-type inequalities in [2], [4] and [6].
The next simple characterisation of (g, ϕ h ) −convex dominated functions holds. (1) f is (g, ϕ h ) −convex dominated on I.
(2) The mappings g − f and g + f are ϕ h − convex on I.
(3) There exist two ϕ h −convex mappings l, k defined on I such that for all x, y ∈ I and t ∈ [0, 1] . The two inequalities may be rearranged as which are eqivalent to the ϕ h −convexity of g + f and g − f, respectively.
2⇐⇒3 Let we define the mappings f, g as f = 1 2 (l − k) and g = 1 2 (l + k). Then if we sum and subtract f and g, respectively, we have g + f = l and g − f = k. By the condition 2 in Lemma 1, the mappings g − f and g + f are ϕ h −convex on I, so l, k are ϕ h −convex mappings on I too.
To prove the inequality in (2.3), firstly we use the Definition 5 for x = a and y = b, we have Then, we integrate the above inequality with respect to t over [0, 1] , we get If we substitute x = tϕ (a) + (1 − t) ϕ (b) and use the fact that So, the proof is completed. Corollary 1. Under the assumptions of Theorem 1 with h (t) = t, t ∈ (0, 1), we have [2]. ≤ g (ϕ (a)) + g (ϕ (b)) s + 1

Remark 1. If function ϕ is the identity in (2.4) and (2.5), then they reduce to Hermite-Hadamard type inequalities for convex dominated functions proved by Dragomir, Pearce and Pečarić in
g (x) dx.