Abstract
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.
1. Introduction
In the last few decades, the classical convexity has a rapid development in fractional calculus [1]. 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
, and
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 [5]:
In the literature, we can notice that Hermite-Hadamarad inequality (2) has been applied to distinct convexities like exponential convexity [6, 7], -convexity [8], quasiconvexity [9, 10], GA-convexity [11], -convexity [12], MT-convexity [13], 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 [21], Ostrowski [22], Rozanova [23], Gagliardo-Nirenberg [24], Olsen [25], Hardy [26], 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. [18] 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 [19]). 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 [14] deduce to the classical RL fractional integrals [9].(ii)If =1, we get fractional integrals of function with respect to function , which is defined by [16, 17]:
Lemma 5. [31] 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
where
for
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.
Remark 7. From Theorem 6, we can get following special case:
If , then inequality (9) becomes
Lemma 8. [31] 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:
Remark 9. From Lemma 8, we obtain the following special case:
If , then equality (24) becomes
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:
Proof. By using Lemma 8 and properties of modulus, we get Since is -convex on for , so So, using (28), we obtain where This completes our proof.
Remark 11. From Theorem 10, we can get following inequalities: (1)If , then inequality (26) becomes (2)If and , then inequality (26) becomes (3)If , and , then inequality (26) becomes
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:
Proof. Since is -convex on for , so By using power mean integral, Lemma 8, and -convexity of , we have where
Remark 13. From Theorem 12, we can get following special cases: (1)If , then inequality (34) becomes (2)If and , then inequality (34) becomes (3)If , and , then inequality (34) becomes
Theorem 14. 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
Proof. Since is -convex on , for , we get By using Hölder’s inequality, Lemma 8, -convexity of , and properties of modulus, we get where This completes the proof.
Remark 15. From Theorem 14, we can obtain following special cases: (1)If , then inequality (41) becomes (2)If and , then inequality (41) becomes (3)If , and , then inequality (41) becomes
3. Conclusion
In this paper, we established Hermite-Hadamarad Fejér type inequalities for -convex function by using weighted fractional integrals. Our results are extensions and generalizations of many existing results in the literature.
Data Availability
All data required for this research is included within the paper.
Conflicts of Interest
The authors of this paper declare that they have no competing interests.
Authors’ Contributions
Lei Chen analyzed the results, Waqas Nazeer proposed the problem, Farman Ali wrote the final version of the paper, Thongchai Botmart verified the results and arranged funding for this paper, and Sarah Mehfooz wrote the first version of the paper. Lei Chen and Farman Ali contributed equally to this work and are first co-authors.