Some Integral Inequalities for 
 n
 -Polynomial 
 ζ
 -Preinvex Functions

In this paper, we study the properties of 
 
 n
 
 -polynomial 
 
 ζ
 
 -preinvex functions and establish some integral inequalities of Hermite-Hadamard type via this class of convex functions. Moreover, we discuss some special cases which provide a significant complement to the integral estimations of preinvex functions. Applications of the obtained results to the inequalities for special means are also considered.


Introduction and Preliminaries
The geometric inequalities involving volume, surface area, mean width, etc. in the Orlicz space have attracted considerable attention of researchers, and the convexity properties of functions have been a powerful tool for dealing with various problems of convex geometry (see [1,2]). This suggests that it is a significant work to develop new inequalities for generalized convex functions. For this purpose, let us start with recalling some concepts and notations on the convexity of functions.
A set C ⊂ ℝ is said to be convex if for any x, y ∈ C and t ∈ ½0, 1.
A function F : C ↦ ℝ is said to be convex if the inequality holds for any x, y ∈ C and t ∈ ½0, 1.
In recent years, the classical concept of convexity has been extended and generalized in different directions. Mititelu [3] introduced the notion of invex set, as follows.
Definition 1 [3]. Let X ⊂ ℝ be a nonempty set and η : ℝ × ℝ ↦ ℝ be a real-valued function. A set X is said to be invex with respect to η if for all x, y ∈ X and t ∈ ½0, 1.
The invexity would reduce to the classical convexity if ηðy, xÞ = y − x. Weir and Mond [4] defined the class of preinvex functions as follows.
Definition 2 [4]. Let X ⊂ ℝ be a nonempty invex set with respect to η : ℝ × ℝ ↦ ℝ. A function F : X ↦ ℝ is said to be preinvex with respect to η if the inequality holds for all x, y ∈ X and t ∈ ½0, 1.
As a generalization of convex functions, Gordji et al. [5] introduced the notion of ζ-convex function.
The properties of convexity have numerous applications in different fields of pure and applied mathematics; especially, the concept of convexity has close relation with the theory of inequalities. Many inequalities are direct consequences of the applications of classical convexity. As is known to us, the Hermite-Hadamard inequality is one of the most significant result associated with convex functions, it reads as follows.
Let F : ½a, b ⊂ ℝ → ℝ be a convex function, then Noor [6] obtained a generalization of classical Hermite-Hadamard's inequality using the class of preinvex functions, as follows.
Recently, Toplu et al. [19] proposed the concept of n -polynomial convex functions and investigated their properties.
In this paper, we shall introduce a new class of n-polynomial convex functions based on a different form of inequality in the definition compared with [19], which is convenient to the generalizations and applications of n-polynomial convexity. More specifically, we will define a class of convex functions called as n-polynomial ζ-preinvex functions. We then show that this class of convex functions contains a number of other classes of convex functions. Furthermore, we establish some new integral inequalities of Hermite-Hadamard type for n-polynomial ζ-preinvex func-tions. Finally, we apply the obtained inequalities to establish two inequalities for special means.
Firstly, we introduce the notion of n-polynomial ζ-preinvex functions.
holds for all a, b ∈ X and t ∈ ½0, 1.
Note that if we take n = 1, then we have 1-polynomial ζ -preinvexity, which is just the ζ-preinvex functions defined by the inequality If we take ζðFðbÞ, FðaÞÞ = FðbÞ − FðaÞ, then we obtain the class of n-polynomial preinvex functions, which is defined by the inequality If we take ηðb, aÞ = b − a, then we get the class of n -polynomial ζ-convex functions, which is defined by the inequality If we take n = 1 in inequality (11), then we have the class of ζ-convex functions. Furthermore, we obtain the classical convex functions by setting ζðFðbÞ, FðaÞÞ = FðbÞ − FðaÞ.
If we take n = 2 in Definition 4, then we have the class of 2 -polynomial ζ-preinvex functions, which is defined by the following inequality: Note that 0 ≤ t ≤ 3t − t 2 /2, this shows that, for every nonnegative bifunction ζ, the ζ-preinvex function is also the 2 -polynomial ζ-preinvex functions. More generally, we have the following result. Journal of Function Spaces Proposition 5. For every nonnegative bifunction ζ and n ≥ 2, if F : X ↦ ℝ is a ðn − 1Þ-polynomial ζ-preinvex function, then F is a n-polynomial ζ-preinvex function.
To verify the validity of Proposition 5, it is enough to show that for any n ≥ 2 and t ∈ ½0, 1. Direct computation gives which implies the required inequality (13).
As a consequence, we obtain the following.

Main Results
In this section, we establish some new Hermite-Hadamardtype inequalities using the class of n-polynomial ζ-preinvex functions. We first need to introduce the notation called Condition C, which was presented by Mohan and Neogy in [20].
Condition C. Let X ⊂ ℝ be an invex set with respect to bifunction ηð:, :Þ, we say that the bifunction ηð:, :Þ satisfies the Condition C, if for any x, y ∈ X and t ∈ ½0, 1, we have Note that for any x, y ∈ X, t 1 , t 2 ∈ ½0, 1 and from Condition C, we can deduce Throughout the paper we assume that Condition C is satisfied for the domain with respect to bifunction ηð:, :Þ as a precondition.
Proof. Using the definition of n-polynomial ζ-preinvex function and Condition C, we have Hence, we obtain Integrating both sides of the above inequality with respect to t on ½0, 1, it follows that 3 Journal of Function Spaces that is, The left-hand side inequality of (17) is proved.
On the other hand, from the definition of n-polynomial ζ -preinvex function, one has Integrating both sides of the above inequality with respect to t on ½0, 1, we obtain This proves the right-hand side inequality of (17). The proof of Theorem 8 is complete.
Before we put forward another kind of integral inequality of Hermite-Hadamard type, we need to prove an auxiliary result, which will play a key role in deducing subsequent results. For the sake of simplicity, we let I = ½a, a + ηðb, aÞ and let I ∘ be the interior of I .
Proof. Let Integrating by parts yields Similarly, Substituting the formulations of I 1 and I 2 in (25) leads to the desired identity (24). The proof of Lemma 9 is complete.
We shall now give some estimations of bounds for Hermite-Hadamard-type inequalities. where Proof. Using Lemma 9 and the assumption that |F ′ | is n -polynomial ζ-preinvex function, we have 4 Journal of Function Spaces which implies the desired inequality (28) since This completes the proof of Theorem 10.
Next, we discuss some special cases of Theorem 10.
We now discuss some special cases of Theorem 11.
Proof. Note that jF ′ j q is n-polynomial ζ-preinvex function, by using the power mean inequality, we have Here, K 1 and K 2 are formulated as that of Theorem 10. This completes the proof of Theorem 12.
We now discuss some special cases of Theorem 12.

Journal of Function Spaces
This completes the proof of Theorem 13.
Let us now discuss some special cases of Theorem 13.

Application to Special Means
Let us recall the definitions of the arithmetic mean, weighted arithmetic mean, and the mean for functions, as follows: (1) The arithmetic mean A a 1 , a 2 , ⋯, a n ð Þ= a 1 + a 2 +⋯+a n n : ð54Þ (2) The weighted arithmetic mean A a 1 , a 2 , ⋯, a n ; p 1 , p 2 , ⋯, p n ð Þ = p 1 a 1 + p 2 a 2 +⋯+p n a n p 1 + p 2 +⋯+p n : (3) The mean of the function Φ on ½a, b We establish the following inequalities for special means.