1. Introduction
Let
be an interval and let
be a continuous function. Then, the function
is called convex if it satisfies
The function is called concave whenever is convex.
For convex functions
, there is an important integral inequality in the literature, namely the Hermite–Hadamard or, briefly, the
integral inequality, which is given by [
1]:
where
belong to
In the literature, one can observe that the
integral inequality (
2) has been applied to different classes of convexity such as
–convexity [
2], quasi-convexity [
3,
4],
s–convexity [
5],
–convexity [
6], exponentially convexity [
7,
8],
–convexity [
9], and the readers can consult [
10,
11] to find other types.
As we know, fractional calculus is a generalized form of integer order calculus. Various forms of fractional derivatives including
, Hadamard, Caputo, Caputo–Hadamard, Riesz,
–
, Prabhakar, and weighted versions [
12,
13,
14,
15,
16] have been developed to date. Most of these versions are described in the
sense based on the corresponding fractional integral. Many integer-order integral inequalities such as Ostrowski [
17], Simpson [
18], Hardy [
19], Olsen [
20], Gagliardo–Nirenberg [
21], Opial [
22,
23] and Rozanova [
24] have been generalized and reformulated from the fractional point of view.
In addition, in 2013, the
integral inequality (
2) was generalized and reformulated by Sarikaya et al. [
25] in terms of
fractional integrals. Their result is given by:
where
is assumed to be a positive convex function, continuous on the closed interval
and for Lebesgue, almost all
when
with
, where
and
are the left- and right-sided
fractional integrals of order
, defined by [
12]:
respectively.
The inequality (
3) is also known as the endpoint
inequality due to using the ends
,
of the interval.
On the other hand, the endpoint
inequality (
3) has been applied for various classes of convexity such as
–convexity [
26],
F–convexity [
27],
–convexity [
28],
–convexity [
29]. The reader can find other types of convexity in the literature, which in particular, is true for [
30]. In the mean time, applying the end-point
inequality to other models of fractional calculus has received a huge amount of attention. For example, this is true for
fractional models [
31], conformable fractional models [
32,
33], generalized fractional models [
34],
fractional models [
35,
36], tempered fractional models [
37], and
- and Prabhakar fractional models [
38].
After extending the important field of the integral inequalities in (
2) and (
3), a new version of the endpoint
inequality (
3) was found by Sarikaya and Yildirim [
39], namely the midpoint
inequality due to using the midpoint
of the interval, which is given by
where the function
is convex and continuous.
Definition 1 ([
40]).
Let be a function. Then, we say g is symmetric with respect to if Based on above definition, in [
41], Fejér found a new extension of the
type inequality (
2), namely the
type inequality, and the result is as follows:
where
g is the integrable function, and Işcan [
42] found the endpoint version of (
7) in the sense of
fractional integrals, which is also the extension of (
3). The result is as follows:
where
is convex and continuous and the function
g belongs to
and is symmetric (see Definition 1).
It is worth mentioning that the midpoint version of (
8) has not been found yet, even though many related inequalities of midpoint type were obtained in [
43].
Recently, Mohammed et al. [
44] found a new endpoint
-inequality in terms of weighted fractional integrals with positive weighted symmetric function in a kernel, and their result is as follows:
Here, is a convex and continuous function, a monotone increasing function from the interval onto itself with a continuous derivative on the open interval and is an integrable function, which is symmetric with respect to where
Definition 2. Let and be an increasing positive and monotone function on the interval with a continuous derivative on the open interval Then, the left-sided and right-sided the weighted fractional integrals of a function according to another function on are defined by [
15]:
for such that . Remark 1. From Definition 2, we can obtain the following special cases.
If and , then the weighted fractional integrals (10) reduce to the classical fractional integrals (4). If , we obtain the fractional integrals of the function with respect to the function , which is defined by [
13,
14]:
In this article, we will investigate the midpoint version of (
9) and some related
inequalities by using the weighted fractional integrals (
10) with positive weighted symmetric functions in the kernel.
The rest of our article is structured in the following way: In
Section 2, we will prove the necessary and auxiliary lemmas, including the midpoint version of (
9). In
Section 3, we will prove our main results, including new midpoint fractional
integral inequalities with some related results. We will present some concluding remarks in
Section 4.