On n-polynomial p-convex functions and some related inequalities

*Correspondence: chuyuming2005@126.com 2Department of Mathematics, Huzhou University, Huzhou, China Full list of author information is available at the end of the article Abstract In this paper, we introduce a new class of convex functions, so-called n-polynomial p-convex functions. We discuss some algebraic properties and present Hermite–Hadamard type inequalities for this generalization. Moreover, we establish some refinements of Hermite–Hadamard type inequalities for this new class.


Introduction
Some geometric properties of convex sets and, to a lesser extent, of convex functions were studied before 1960 by outstanding mathematicians, first of all by Hermann Minkowski and Werner Fenchel. At the beginning of 1960 convex analysis was greatly developed in the works of R. Tyrrell Rockafellar and Jean-Jacques Morreau who initiated a systematic study of this new field. There are several books devoted to different aspects of convex analysis and optimization. See [1][2][3][4][5][6].
The idea of convexity is not new one even it occurs in some other form in Archimede's treatment of orbit length. Nowadays, the application of several works on convexity can be directly or indirectly seen in various subjects like real analysis, functional analysis, linear algebra, and geometry. Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences. Many articles have been written by a number of mathematicians on convex functions and inequalities for their different classes. In the last few decades, the subject of convex analysis got rapid development because of its geometry and its role in the optimization. The deep relation between convex analysis and fractional calculus can never be ignored. For recent work on fractional calculus, we refer to [13][14][15][16][17].
Let ψ : I → R be a convex function, then for all x, y ∈ I and t ∈ (0, 1), the following holds: For the extended version of the above inequality, see [18,19].
In [20], Lipot Fejér presented an extended version of (1.2) inequality known as Fejér inequality or a weighted version of the Hermite-Hadamard inequality. If ψ : I → R is a convex function, then where c ≤ d, and w : I → R is nonnegative, integrable, and symmetric about c+d 2 . The present paper is organized as follows: First we give some preliminary material and basic definition for n-polynomial p-convex function. In the second section we give some basic results for our newly defined generalization. Next we develop Hermite-Hadamard type inequality. In the last section, we give some theorems related to our work.

Preliminaries
We start with some basic definitions. It can be easily seen that, for p = 1, p-convexity is reduced to the classical convexity of functions defined on I ⊂ (0, ∞). Now we recall the definition of harmonically convex function as follows. [23]) Let I ⊂ R be an interval. Then a realvalued function ψ : I → R is said to be harmonically convex if

Definition 2.3 (Harmonic convex function
holds for all x, y ∈ I and t ∈ [0, 1]. In [24] n-polynomial convexity has been defined. Definition 2.4 (n-polynomial convex function) Let n ∈ N. A nonnegative function ψ : I → R is called n-polynomial convex function if, for every x, y ∈ I and t ∈ [0, 1], We will denote by POLC(I) the class of all n-polynomial convex functions on interval I.
We note that every n-polynomial convex function is an h-convex function with the function In [25] n-polynomial harmonically convexity has been defined. Definition 2.5 (n-polynomial harmonic convex function) Let n ∈ N. A nonnegative function ψ : I → R is called n-polynomial harmonically convex function if, for every x, y ∈ I and t ∈ [0, 1], From Definition 2.5, for n = 2, we can see that the class of n-polynomial harmonically convex functions satisfies the inequality for all x, y ∈ I and t ∈ [0, 1]. Now we are going to introduce a new generalization of n-polynomial convex function.

Basic results
In this section we derive some basic results and propositions related to our new generalization.
The following proposition shows the linearity of n-polynomial p-convex function.
Proposition 3.1 Let φ : I → R be a nonnegative n-polynomial p-convex function, and where for n ∈ N, x, y ∈ I, p > 0 and t ∈ [0, 1], then ψ + φ is an n-polynomial p-convex function.
Proof Let ψ and φ be two n-polynomial p-convex functions, then for all x, y ∈ I, p > 0 and ∈ [0, 1] we have this assures the n-polynomial p-convexity of ψ + φ.
Now we will discus the scalar multiplication of n-polynomial p-convex function.
Proposition 3.2 Let ψ : I → R be a nonnegative n-polynomial p-convex function and λ > 0, where for n ∈ N, x, y ∈ I, p > 0 and t ∈ [0, 1], then λψ : I → R is also an n-polynomial p-convex function.
Proof Let ψ be an n-polynomial p-convex function, then for all x, y ∈ I, p > 0 and t ∈ [0, 1], where λ > 0, we have which shows that λψ is also an n-polynomial p-convex function.
is also n polynomial p-convex function.
Proof Fix x, y ∈ R n , p > 0 and t ∈ [0, 1], then for every i ∈ I we have

Hermite-Hadamard type inequality for n-polynomial p-convex function
The goal of this paper is to establish some inequalities of Hermite-Hadamard type for n-polynomial p-convex function.
Proof Fix x, y ∈ R n , p > 0, and t ∈ [0, 1], then for every i ∈ I, by the definition of npolynomial p-convex function of ψ,we have Integration in the last inequality with respect to t ∈ [0, 1] yields that After solving the above inequality (4.3), we get (4.5) which is the left-hand side of the theorem.
To prove the right-hand side of the theorem, take since ψ is an n-polynomial p-convex function: which is the right-hand side of the theorem. 2. For p = -1, we obtain Hermite-Hadamard type inequality for n-polynomial harmonically convex function [25].

New inequalities for n-polynomial p-convex function
In this section, we establish new estimates that refine Hermite-Hadamard inequality for a function whose first derivative is absolute value, raised to a certain power which is greater than one.
In [26] the following lemma is given, which will be helpful for generating refinements of Hermite-Hadamard type inequality. and Proof The definition of n-polynomial convexity and Lemma 5.1 yields the following: We get This completes the proof.

Corollary 5.4
If we take n = 1 and p = 1 in inequality (4.1), we get the following inequality: This inequality coincides with the inequality in [26].
In [30], Iscan gave a refinement of Holder integral inequality as follows.
which is an n-polynomial p-convex function of |ψ | q , we get

Corollary 5.9
If we take n = 1 and p = 1 in inequality (4.1), we get the following inequality: This inequality coincides with the inequality in [26].  1t s ψ (d) This completes the proof of the theorem.

Corollary 5.12
If we take n = 1 and p = 1 in (4.1), we get the following inequality: This inequality coincides with the inequality in [26] with q = 1.