Generalized Fractional Hadamard and Fejér–Hadamard Inequalities for Generalized Harmonically Convex Functions

Abstract

In this paper, we define a new function, namely, harmonically -convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically -convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically -convex, harmonically -convex, and related functions and for already known fractional integral operators.

1. Introduction and Preliminary Results

Fractional integral inequalities are the generalizations and extensions of classical integral inequalities using fractional integral/derivative operators. There are many well-known inequalities which have been extended for fractional calculus operators: for example, Hadamard, Minkowski, Ostrowski, Grüss, Ostrowski–Grüss, and Chebyshev inequalities have been extensively studied in recent decades, see [1–8]. The aim of this paper is to present the Hadamard and the Fejér–Hadamard inequalities for fractional integral operators for harmonically -convex functions. We begin from the fractional integral operators defined by Andrić et al. in [2] containing an extended generalized Mittag-Leffler function in their kernels.

Definition 1. (see [2]). Let , , and with , , and . Let and . Then, the generalized fractional integral operators and are defined as follows:whereis the extended generalized Mittag-Leffler function.
In [9], Farid defined elegantly the unified integral operators as follows:

Definition 2. Let , , be the functions such that is positive and and is differentiable and strictly increasing. Also, let be an increasing function on and , , and with , , and . Then, for , the integral operators and are defined as follows:The following definition of extended generalized fractional integral operators containing an extended Mittag-Leffler function in the kernel can be extracted by taking in Definition 2.

Definition 3. Let , , be the functions such that is positive and and is differentiable and strictly increasing. Also, let , , and with , , and . Then, for , the generalized fractional integral operators are defined as follows:Fractional integral operators (4) and (5) produce particularly various well-known fractional integral operators.
The Hadamard inequality gives the bounds of integral mean of a convex function on a closed interval. It is equivalently defined by the convex function. Convex functions have a significant role in the field of mathematical inequalities.

Definition 4. A function is said to be convex ifholds for all and .
The Hadamard inequality is stated in the following theorem.

Theorem 1. Let be a convex function. Then, the following inequality holds:

The Fejér–Hadamard inequality proved by Fejér in [10] generalizes the Hadamard inequality, and it is given as follows:

Theorem 2. Let be a convex function and be nonnegative, integrable, and symmetric about . Then, the following inequality holds:

The definition of the convex function motivates the researchers to define new notions and their consequences. A notion of the harmonically convex function is defined in [11].

Definition 5. Let be an interval such that ; a function is said to be harmonically convex ifholds for all and .
This notation is further extended to the harmonically -convex function in [12] and harmonically -convex function in [13].

Definition 6. (see [12]). A function , , is said to be harmonically -convex if is nonnegative andholds for all , , , and .

Definition 7. (see [13]). Let be a nonnegative function. A function is said to be harmonically -convex ifholds for all and .
In the upcoming section, we define a new generalized notion, namely, harmonically -convex function, which unifies all known definitions of harmonically convex functions. By using this new definition, we will prove Hadamard and Fejér–Hadamard inequalities for generalized fractional integral operators involving an extended generalized Mittag-Leffler function. Also, results for harmonically -convex functions and harmonically -convex functions are deduced.

2. Main Results

First, we define the harmonically -convex function as follows:

Definition 8. Let be a nonnegative function. A function is said to be harmonically -convex ifholds for all , , , and .
It unifies the definitions of harmonically -convexity and harmonically -convexity of functions. For different specific choices of , almost all kinds of well-known harmonically convex functions can be obtained, see the remark as follows.

Remark 1.  (i)If , then a harmonically -convex function can be obtained [12](ii)If and , then a harmonically -convex function can be obtained [14](iii)If and , then a harmonically -convex function can be obtained [14](iv)If , then a harmonically -function can be obtained [13](v)If , , and , then a harmonically -convex function can be obtained [13](vi)If , , and , then a harmonically Godunova–Levin function can be obtained [13](vii)If , , and , then a harmonically -Godunova–Levin function can be obtained [13](viii)If we set and , then a harmonically -convex function can be achieved [13](ix)By putting , , and , then a harmonically convex function can be obtained [11]We give another new notion of the harmonically -convex function by setting in Definition 8 as follows.

Definition 9. Let be a nonnegative function. A function is said to be harmonically -convex ifNow, we apply Definition 8 and operators (4) and (5) to find the Hadamard and Fejér–Hadamard inequalities.
In the whole paper, we have used the following notations frequently:

Theorem 3. Let , , range , be the functions such that is positive and and is differentiable and strictly increasing. If is harmonically -convex on , then for operators (4) and (5), we havewhere and for all .

Proof. Since is harmonically -convex, for all , the following inequality holds:By setting and in (16), we haveMultiplying (17) by and integrating over , we getBy setting and in (18) and using (4) and (5), the first inequality of (15) can be obtained. On the contrary, from the harmonically -convexity of , we haveMultiplying (19) by and then integrating over , we getBy setting and in (20) and using (4) and (5), the second inequality of (15) can be obtained.

Remark 2.  (i)By setting , , , and , Theorem 3.1 of [15] is obtained(ii)By setting , , and , Theorem 2.1 of [16] is obtained(iii)By setting , , , and , Theorem 4 of [6] is obtained

Theorem 4. Let , , range , range , be the functions such that is positive and , is differentiable and strictly increasing, and is nonnegative and integrable. If is harmonically -convex and , then for operators (4) and (5), we havewhere and for all .

Proof. Multiplying (17) by and then integrating over , we getBy choosing , that is, , in (22) and using (4) and (5) and the condition , the first inequality of (21) can be obtained. For the second inequality of (21), multiplying (19) by and then integrating over , we getBy setting in (23) and using (4) and (5) and the condition , the second inequality of (21) can be obtained. □

Remark 3.  (i)By setting , , , and , Theorem 3.1 of [15] is obtained(ii)By setting , , , and , Theorem 2.1 of [16] is obtained(iii)By setting , , , , and , Theorem 4 of [6] is obtained(iv)By setting , , , , and , Theorem 8 of [4] is obtained(v)By setting , , , , , and , Theorem 4 of [17] is obtained

Theorem 5. Let , , range , be the functions such that is positive and and is differentiable and strictly increasing. If is harmonically -convex on , then for operators (4) and (5), we havewhere and .

Proof. By setting and in (16), we haveMultiplying (25) by and then integrating over , we getBy setting and in (26) and using (4) and (5), the first inequality of (24) can be obtained. On the contrary, from the harmonically -convexity of , we haveMultiplying (27) by and integrating over , we getBy setting and in (28) and using (4) and (5), the second inequality of (24) can be obtained.

Remark 4.  (i)By setting , , , and , Theorem 3.3 of [15] is obtained(ii)By setting , , and , Theorem 2.3 of [16] is obtained

Theorem 6. Let , , range , range , be the functions such that is positive and , is differentiable and strictly increasing, and is nonnegative and integrable. If is harmonically convex and , then for operators (4) and (5), we havewhere and for all .

Proof. Multiplying (25) by and integrating over , we getBy choosing in (30) and using (4) and (5) and the condition , the first inequality of (29) can be obtained. On the contrary, by the harmonically -convexity of , multiplying (27) by and integrating over , we getBy choosing in (31) and using (4) and (5) and the condition , the second inequality of (29) can be obtained.

Remark 5.  (i)By setting , , , and , Theorem 3.6 of [15] is obtained(ii)By setting , , and , Theorem 5 of [16] is obtained

2.1. Results for Harmonically -Convex Functions

By setting in Theorems 3–6, the results for harmonically -convex functions are obtained as follows.

Theorem 7. Under the assumptions of Theorem 3, the following inequality holds for harmonically -convex functions:

Theorem 8. Under the assumptions of Theorem 4, the following inequality holds for harmonically -convex functions:

Theorem 9. Under the assumptions of Theorem 5, the following inequality holds for harmonically -convex functions:

Theorem 10. Under the assumptions of Theorem 6, the following inequality holds for harmonically -convex functions:

2.2. Results for Harmonically -Convex Functions

By setting in Theorems 3–6, the results for harmonically -convex functions are obtained as follows.

Theorem 11. Under the assumptions of Theorem 3, the following inequality holds for harmonically -convex functions:

Theorem 12. Under the assumptions of Theorem 4, the following inequality holds for harmonically -convex functions:

Theorem 13. Under the assumptions of Theorem 5, the following inequality holds for harmonically -convex functions:

Theorem 14. Under the assumptions of Theorem 6, the following inequality holds for harmonically -convex functions:

3. Concluding Remarks

The presented work includes two generalized fractional versions of the Hadamard inequality (stated in Theorems 3 and5) as well as two generalized fractional versions of the Fejér–Hadamard inequality (stated in Theorems 4 and6) for a new generalized harmonic convexity. Their connections with already published results are shown in Remarks 2, 3, 4, and 5. Also, we work out these inequalities for harmonically -convex and -convex functions. The reader can obtain a series of Hadamard and Fejér–Hadamard inequalities for fractional integral operators containing Mittag-Leffler functions for several kinds of convex functions.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).