Hermite–Hadamard type inequalities for differentiable hφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_{\varphi}}$$\end{document}-preinvex functions

In this paper, we introduce a new class of convex functions which is called hφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_{\varphi}}$$\end{document}-preinvex functions. We prove several Hermite–Hadamard inequalities for hφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h_{\varphi}}$$\end{document}-preinvex functions. Some special cases are also discussed. Results proved in this paper continue to hold for these special cases. Our results may stimulate further investigation regarding variant forms of Hermite–Hadamard type inequalities.


Introduction
Let f : I ⊆ R → R be a convex function with a < b and a, b ∈ I . Then the following double inequality is known as Hermite-Hadamard inequality in the literature and showed that ϕ-convex functions are nonconvex functions. Noor [12] extended Hermite-Hadamard type inequalities for ϕ-convex functions. Motivated by the ongoing research on generalizations and extensions of classical convexity, Varosanec [24] introduced the class of h-convex functions. She studied the basic properties of h-convex functions and noticed that along with classical convexity this class also generalizes several other classes of convex functions such as, s-convex functions [2], Q functions [6] and P functions [5] respectively. Noor et al. [16] introduced another generalization of classical convexity which is called as h ϕ -convex functions. Several Hermite-Hadamard type inequalities are proved for h ϕ -convex functions.
Motivated and inspired by the recent research in this field, we introduce and consider a new class of convex functions, which is called h ϕ -preinvex functions. This class includes several known and new classes of convex functions such as ϕ-convex functions [13], preinvex functions [25], h-convex functions [24] as special cases. We derive several new Hermite-Hadamard type integral inequalities for h ϕ -preinvex functions and their variant forms. Results obtained in this paper continue to hold for these special cases. Our results represent significant refinement and improvement of the previous results. The interested readers are encouraged to find novel and innovative applications of h ϕ -preinvex functions.

Preliminaries
In this section, we define and recall some basic concepts and results. Let R n be the finite dimensional Euclidian space. Also let 0 ≤ ϕ ≤ π 2 be a continuous function.
The ϕ-invex set K ϕη is also called ϕη-connected set. Note that the convex set with ϕ = 0 and η(v, u) = v − u is a ϕ-invex set, but the converse is not true (see [15]).
Remark 2.2 Definition 2.1 says that there is a path starting from a point u which is contained in K ϕη . We do not require that the point v should be one of the end point of the path. Note that, if we demand that v should be an end point of the path for every pair of points u, v ∈ K ϕη , then e iϕ η(v, u) = v − u, if and only if, e iϕ = 1 and η(v, u) = v − u. Then the ϕ-invex K ϕη becomes the convex set K . It is clear that every convex set is a ϕ-invex set, but the converse is not necessarily true.

Definition 2.3 ([13])
Let K ϕ be a set in R n . Then the set K ϕ is said to be ϕ-convex with respect to ϕ, if and only if For ϕ = 0, the set K ϕ reduces to the classical convex set K . That is,

Definition 2.4 ([25])
A set K η is said to be invex set with respect to bifunction η(., .), if The invex set K η is also called η-connected set.

Definition 2.5
Let h : J ⊆ R → R be a nonnegative function. A function f on the set K ϕη is said to be h ϕ -preinvex function with respect to ϕ and bifunction η, if Remark 2.6 One can deduce several known concepts from Definition 2.5 as: (1) For h(t) = t Definition 2.5 reduces to the definition for ϕ-preinvex functions (see [15]).
(4) For ϕ = 0 and η(v, u) = v − u Definition 2.5 reduces to the definition for h-convex functions (see [24]). Now we discuss some special cases of Definition 2.5. (2) we have the definition for s ϕ -preinvex functions.

Definition 2.7
A function f on the set K ϕη is said to be s ϕ -preinvex function with respect to ϕ and η, if

II.
For h(t) = 1 in (2) we have the definition for P ϕ -preinvex functions.

Definition 2.8
A function f on the set K ϕη is said to be P ϕ -preinvex function with respect to ϕ and η, if From the above discussion, it is obvious that the class of h ϕ -preinvex functions is quite general and unifying one. This is one of the main motivation of this paper.

Main results
In this section, we will discuss our main results. Using the technique of [1], we prove the following Lemma which play a key role in our results.

Lemma 3.1 Let I ⊆ R be a open invex set with respect to bifunction
This completes the proof.

Theorem 3.2 Let I ⊆ R be a open invex set with respect to bifunction
This completes the proof.

III.
If h(t) = 1, then we have the result for P ϕ -preinvexity.

Theorem 3.6 Let I ⊆ R be a open invex set with respect to
Proof Using Lemma 3.1, we have Now using wel-known Holders inequality, we have This completes the proof.
Now we have some special cases.

II.
If h(t) = t s , then we have the result for s ϕ -preinvexity. + e iϕ η(b, a)]. Let | f | p be s ϕ -preinvex on I , 1

III.
If h(t) = 1, then we have the result for P ϕ -preinvexity.  L 1 [a, a + e iϕ η(b, a)]. Let | f | p be P ϕ -preinvex on I , 1 Theorem 3.10 Let I ⊆ R be a open invex set with respect to η : I × I → R. Suppose that f : I → R is a differentiable function such that f ∈ L 1 [a, a + e iϕ η(b, a)]. If | f | q , q > 1 is h ϕ -preinvex on I , then, for  η(b, a) > 0, Using power-mean inequality, we have This completes the proof.  η(b, a)]. If | f | q , q > 1 is ϕ-preinvex on I , then, for

II.
If h(t) = t s , then we have the result for s ϕ -preinvexity.

III.
If h(t) = 1, then we have the result for P ϕ -preinvexity.  η(b, a)]. If | f | q , q > 1 is P ϕ -preinvex on I , then, for Using the technique of [1], we prove the following result which helps us in proving our next results.