Hermite-Hadamard-Fejér Inequality Related to Generalized Convex Functions via Fractional Integrals

Abstract

This paper deals with Hermite-Hadamard-Fejér inequality for -convex functions via fractional integrals. Some mid-point and trapezoid type inequalities related to Hermite-Hadamard inequality when the absolute value of derivative of considered function is -convex functions are obtained. Furthermore, a refinement for classic Hermite-Hadamard inequality via fractional integrals is given when a positive -convex function is increasing.

1. Introduction and Preliminaries

In 1906, L. Fejér [1] proved the following integral inequalities which are known in the literature as Fejér inequality:where is convex and is integrable and symmetric to .

If we consider in (1) that , then we recapture the known Hermite-Hadamard inequality.

On the other hand, the concept of -convex has been introduced in [2, 3] as a generalization of preinvex functions [4–6] and -convex functions [7–9]. We remind these introductory concepts in the following with some geometric interpretation.

(Preinvex Functions)

Definition 1. A set is invex with respect to a real bifunction , if If is an invex set with respect to , then a function is said to be preinvex if and implies

In fact, in an invex set for any , there is a path starting from to which is contained in . The point is not necessarily the end point of the path. If for every we need that should be an end point of the path, then reduces to a convex set.

(-Convex Functions)

Definition 2. Consider a convex set and a bifunction . A function is called convex with respect to (briefly -convex), if for all and .

Geometrically, it says that if a function is -convex on , then, for any , its graph is on or under the path starting from and ending at . If should be the end point of the path for every , then we have and the function reduces to a convex one.

(()-Convex Functions)

Definition 3. Let be an invex set with respect to . Consider and . The function is said to be -convex if for all and .

Remark 4. An -convex function reduces to(i)a -convex function if we consider for all ,(ii)a preinvex function if we consider for all ,(iii)a convex function if satisfying (i) and (ii).

Example 5 (see [2]). Consider the function by Define two bifunctions and by and Then is an -convex function. But is not preinvex with respect to and it is not convex (consider , , and ).

Also we need the following short preliminaries about the fractional calculus theory which are used throughout the paper.

Definition 6 (see [10]). Consider . The Riemann-Liouville integrals and of order with are defined by and respectively, where is Gamma function and .

Fejér inequality for convex functions related to fractional integrals has been obtained in [11] as the following theorem.

Theorem 7. Let be convex function with and . If is nonnegative, integrable, and symmetric to , then the following inequalities hold for fractional integrals: with .

Motivated by above works and results, in this paper, we obtain Fejér inequality for -convex functions via fractional integrals. Also we give some mid-point and trapezoid-type inequalities related to Hermite-Hadamard inequality when the absolute value of the derivative of the considered function is -convex functions. Furthermore, we prove that when a positive -convex function is increasing, there exists a refinement for classic Hermite-Hadamard inequality via fractional integrals.

2. Fejér Inequality

In this section, we obtain -convex version of the Fejér inequality related to fractional integrals. For convenience, we separate this inequality to the left and right.

Theorem 8 (Fejér’s left inequality). Let be an invex set with respect to such thatfor all and . Also let be an -convex function, where is an integrable bifunction on . For any with , suppose that and the function is integrable and symmetric to . Then, for , the following inequality holds:

Proof. Using condition (14) and the -convexity of , we haveand with the same argument as above we haveBy the use of Definition 6 and two changes of variable and in (16) and (17), respectively, we obtain the following inequalities:The simple form of (20) along with the fact that is symmetric to leads to the following relations:Also with the same argument as above we have Now adding to implies the result.

To obtain the right part of the Fejér inequality related to fractional integrals, we need a primary lemma.

Lemma 9. Let be an invex set with respect to and with . If is integrable and symmetric to , then

Proof. Since is symmetric to , we have for all . Then

Theorem 10 (Fejér’s right inequality). Let be an invex set with respect to and let be an -convex function, where is an integrable bifunction on . For any with , suppose that the function is integrable and symmetric to and . Then, for , the following inequality holds:

Proof. From -convexity of , using the changes of variables and , respectively, we obtain the two following inequalities:andNow adding (26) to (27) with the fact that is symmetric to implies that Now by the use of Lemma 9 we have which implies the respected inequality.

Corollary 11. If in Theorems 8 and 10 we consider
(i) , then the following inequality holds, which is the classical form of Fejér inequality related to -convex functions:(ii) , then we get Hermite-Hadamard inequality for -convex functions as follows: which is a generalization of inequality in [12].

Corollary 12. If in Theorems 8 and 10 we set for all , then we obtain Fejér inequality for fractional integrals related to -convex functions.

Corollary 13. If in Theorems 8 and 10 we set for all , then we obtain Fejér inequality for fractional integrals related to preinvex functions.

Corollary 14 (see [11]). With all conditions of Corollaries 12 and 13, we have the classic Fejér inequality for fractional integrals.

Corollary 15 (see [12]). If in (34) we consider , then we recapture Hermite-Hadamard inequality for fractional integrals in convex case.

3. Mid-Point and Trapezoid-Type Inequalities

In this section, we obtain, respectively, the mid-point and trapezoid-type inequalities related to (31) when the absolute value of the derivative of the considered function is -convex. In fact, by mid-point-type inequality we mean estimating the difference between left and middle parts of (31) and by trapezoid-type inequality we mean estimating the difference between right and middle parts of (31).

The following lemma is generalization of Lemma 1 obtained in [13] to the preinvex case.

Lemma 16. Let be an open invex set with respect to and let be a differentiable function. For any with , if , then the following equality for fractional integrals holds: where

Proof. Integrating by parts in implies that Similarly, we have Now, by adding all of above equalities, we get to the desired result.

The mid-point-type inequality related to (31) is obtained in the following.

Theorem 17. Let be an open invex set with respect to and let be a differentiable function. Suppose that is an -convex function on and, for any with , . Then for .

Proof. Using Lemma 16, we get Now, using -convexity of , we obtain Analogously, Also, using the fact that for all and , we have and Now adding all of above inequalities implies the required result.

Corollary 18. If in Theorem 17 we consider for all , then Furthermore, if we set for all , then we recapture Theorem 2 in [13].

The following result has been obtained in [14].

Lemma 19. Let be an open invex set with respect to . Also, suppose that is a differentiable function. For any with , if , then the following equality for fractional integral holds:

Now we give the trapezoid-type inequality related to (31).

Theorem 20. Let be an open invex set with respect to and let be a differentiable function. Suppose that is an -convex function on and, for any with , . Then the following inequality for fractional integrals holds:

Proof. Using Lemma 19 and -convexity of , we get where the last equality can be obtained after some calculations in corresponding integrals and utilizing them.

Corollary 21. If in Theorem 20 we consider
(i) for all , then we obtain (ii) for all , then we have Theorem 2.5 in [14].
(iii) conditions of (i) and (ii) together, then we recapture Theorem 3 in [12].

As the last result by using Theorem 2 in [15], we obtain a refinement of Hermite-Hadamard inequality in connection with fractional integrals related to the increasing -convex functions.

Theorem 22 (see [15]). If and are positive increasing functions on , then

Also if and are positive decreasing functions on and is an upper bound for and , then and are positive increasing functions and we have which gives again

Theorem 23. Let be an invex set with respect to . Also let be an increasing positive -convex function. If, for each with , and is integrable on , then, for , the following inequalities hold:

Proof. From -convexity of , we have, and By adding these inequalities, we getMultiplying both sides of (59) by and integrating the resulting inequality with respect to over and using Theorem 22, we obtain Using the changes of variables and , respectively, in above integrals, we have Since and we get Furthermore, since is -convex, then which completes the proof.

Corollary 24. Let be an increasing positive convex function. Then, for , the following inequalities hold:

4. Conclusions

The convexity of a function is the basis for many inequalities in mathematics. Note that, in new problems related to the convexity, generalized notions about convex functions are required to obtain applicable results. One of these generalizations is the notion of -convex functions which can generalize many inequalities related to convex functions such as Hermite-Hadamard inequality, Fejér inequality, and trapezoid-type and mid-point-type inequalities.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

M. De La Sen is very grateful to the Spanish Government for its support by the European Regional Development Fund (ERDF) through Grant DPI2015-64766-R and to UPV/EHU for its support by Grant PGC 17/33.