Advances in Fractional Functional AnalysisView this Special Issue
Some Midpoint Inequalities for -Convex Function via Weighted Fractional Integrals
In this research, by using a weighted fractional integral, we establish a midpoint version of Hermite-Hadamrad Fejér type inequality for -convex function on a specific interval. To confirm the validity, we considered some special cases of our results and relate them with already existing results. It can be observed that several existing results are special cases of our presented results.
In the last few decades, the classical convexity has a rapid development in fractional calculus . We can say that convexity plays a vital role in fractional integral inequalities because of its geometric features [2–4].
Take a function : be a continuous function. Then, this function is called convex if
There are many integral inequalities in the literature and one of the most common inequality is Hermite-Hadamarad or, shortly, the HH integral inequality, which is introduced by :
In the literature, we can notice that Hermite-Hadamarad inequality (2) has been applied to distinct convexities like exponential convexity [6, 7], -convexity , quasiconvexity [9, 10], GA-convexity , -convexity , MT-convexity , and also, other types of convexity (see [14, 15]). Different forms of fractional integrals like Riemann-Liouville (RL), Caputo Fabrizio, Hadamrad, Riesz, Prabhakar, -RL, and weighted integrals [16–20] have been established. A lot of integer-order integral inequalities like Simpson , Ostrowski , Rozanova , Gagliardo-Nirenberg , Olsen , Hardy , Opial [27, 28], and Akdemir et al. [29, 30] have been developed and generalized from fractional point of view.
Definition 1. Let be an interval and be a continuous function. Then, the function is called -convex if
Definition 2.  Let is positive convex function, continuous on closed interval and when with , where left- and right-side RL fractional integrals are defined by where is famous Gamma function and for any positive integer
Definition 3 (see ). Let , and be monotonically increasing positive function with a continuous derivative on Then, the left-sided and the right-sided weighted fractional integrals of according to on are defined by:
In this research, we denote and the inverse of function by .
Remark 4. From Definition 3, we can see some special cases: (i)If and , then weighted fractional integrals  deduce to the classical RL fractional integrals .(ii)If =1, we get fractional integrals of function with respect to function , which is defined by [16, 17]:
Lemma 5.  Assume that is integrable function and symmetric with respect to . Then, (i)For each , we have (ii)For , we have
2. Main Results
Theorem 6. Let and be an -convex function and be an integrable, positive and weighted symmetric function with respect to . If, in addition, is an increasing and positive function from onto itself such that its derivative is continues on then for , the following inequalities are valid:
Proof. The -convexity of on for all gives
setting and Multiplying both sides of inequality (11) by and integrating over we get
From the left side of inequality (12), we use
where . It follows that
By evaluating the weighted fractional operators, we see that
Setting and , one can deduce that By using (14) and (18) in (12), we get The left side of Theorem 6 is completed.
Now, we will prove right side of inequality (9) by using -convexity. Multiply Equation (20) by and integrate over leads us to By using (7) and (14) in (21), we get This completes our proof.
Lemma 8.  Let and be a continuous with a derivative such that and let be an integrable, positive, and weighted symmetric function with respect to . If is a continuous increasing mapping form the interval onto itself with a derivative which is continuous on , then for , the following equality is valid:
Theorem 10. Let and be a differentiable function on the interval such that and let be an integrable, positive, and weighted symmetric function with respect to . If, in addition, is convex on , and is an increasing and positive function from onto itself such that its derivative is continuous on , then for , the following inequalities hold:
Theorem 12. Let and be a continuously differentiable function on the interval such that , and let be integrable, positive, and weighted symmetric function with respect to . If, in addition, is convex on , and is increasing and positive function from onto itself such that its derivative is continuous on , then for , we have: