Some Hermite-Hadamard type inequalities for (P ,m)-function and quasi m-convex functions

Article History: Received 14 February 2019 Accepted 16 May 2019 Available 16 January 2020 In this paper, we introduce a new class of functions called as (P,m)-function and quasi-m-convex function. Some inequalities of Hadamard’s type for these functions are given. Some special cases are discussed. Results represent significant refinement and improvement of the previous results. We should especially mention that the definition of (P,m)-function and quasi-m-convexity are given for the first time in the literature and moreover, the results obtained in special cases coincide with the well-known results in the literature.


Preliminaries
Inequalities present an attractive and active field of research. In recent years, various inequalities for convex functions and their variant forms are being developed using innovative techniques. For some inequalities, generalizations and applications concerning convexity see [1,2]. Recently, in the literature there are so many papers about Pfunction, quasi-convex and m-convex functions. Many papers have been written by a number of mathematicians concerning inequalities for Pfunction, quasi-convex functions and m-convex functions see for instance the recent papers [3][4][5][6][7][8] and the references within these papers. Let ω : I ⊆ R → R be a convex function defined on the interval I of real numbers and λ, µ ∈ I with λ < µ. The following inequality holds.
The inequality (1) is known as Hermite-Hadamard (H-H) integral inequality for convex functions in the literature.
Some refinements of the H-H inequality on convex functions have been extensively studied by researchers (e.g., [1,9]) and the researchers obtained a new refinement of the H-H inequality for convex functions.
Definition 3. A function ω : I ⊆ R → R is said to be quasi-convex if the inequality holds for all λ, µ ∈ I and t ∈ [0, 1]. Denote by QC(I) the set of the quasi-convex functions on the interval I.
We will denote by P (I) the set of P -function on the interval I. Note that P (I) contain all nonnegative quasi-convex and convex functions.
In [10], Dragomir et al. proved the following inequality of Hadamard type for class of Pfunctions.

Some new definitions and their properties
In this section, we will define the (P, m) and quasi-m-convex function supply several properties of this kind of functions.
It is clear that quasi-convexity obtained in quasim-convexity for m = 1.
Proof. Let t ∈ [0, 1] and λ, µ ∈ J be arbitrary. Then This shows simultaneously that J is an interval since it contains every point between any two of its points and ω is a (P, m)-function on the interval J. Proof. For λ, µ ∈ I and t ∈ [0, 1],

Hermite-Hadamard integral inequality for (P , m)-function and quasi-m-convex functions
The main purpose of this paper is to develop concepts of the (P, m)-function and quasi-m-convex functions and to obtain some inequalities of H-H type for these classes of functions.

Remark 2.
Under the conditions of Theorem 9, if m = 1 then, the following inequality holds: The above inequality is the right hand side of the inequality 2.
By a similar argument, if we take From (6) and (8), we obtain Integrating over t ∈ [0, 1], we obtain As it is easy to see that  (10) we deduce the desired result, namely, the inequality (9).
Remark 5. For m = 1, (9) exactly becomes the right hand side of the inequality 2.