A Study of Uniform Harmonic -Convex Functions with respect to Hermite-Hadamard’s Inequality and Its Caputo-Fabrizio Fractional Analogue and Applications

Abstract

In this paper, we introduce the notion of uniform harmonic -convex functions. We show that this class relates several other unrelated classes of uniform harmonic convex functions. We derive a new version of Hermite-Hadamard’s inequality and its fractional analogue. We also derive a new fractional integral identity using Caputo-Fabrizio fractional integrals. Utilizing this integral identity as an auxiliary result, we obtain new fractional Dragomir-Agarwal type of inequalities involving differentiable uniform harmonic -convex functions. We discuss numerous new special cases which show that our results are quite unifying. Finally, in order to show the significance of the main results, we discuss some applications to means of positive real numbers.

1. Preliminaries

In recent years, the classical concepts of convexity have been extended and generalized in different directions using novel and innovative ideas. It has also been observed that these new extensions and generalizations enjoy some nice properties which the classical concepts of convexity have. Everyone is familiar with the fact that convexity has a close relationship with the theory of inequalities. The classical concepts of convexity have played a significant role in the development of the theory of inequalities. Many famously known results in the theory of inequalities can easily be obtained using the convexity property of the functions. Thus, this becomes an interesting problem for research, of obtaining new versions of classical inequalities using new generalizations of convexity. Interested readers may find some useful details regarding convexity, its generalizations, and associated results in [1]. In 2014, Işcan [2] introduced and investigated the notion of harmonic convex functions. He defined the class of harmonic convex functions as follows.

Definition 1. A function is said to be harmonically convex, if holds for all and .

He also obtained a new variant of Hermite-Hadamard’s inequality pertaining to harmonic convex functions as follows.

Theorem 2. Let be a harmonically convex function, and then

Having inspiration from the research work of Sarikaya et al. [3], Işcan and Wu [4] obtained the fractional version of Hermite-Hadamard’s inequality, essentially utilizing the concept of harmonic convex functions. For some recent investigations on harmonic convex functions, see [5, 6].

In [7], the class of uniformly convex functions was defined as follows.

Definition 3. Let be a convex set. A function is said to be uniformly convex with modulus if is increasing, vanishes only at , and

The main motivation of this paper is to introduce the class of uniform harmonic -convex functions. We show that this class contains some other new classes of uniform harmonic convex functions. Using the class of uniform harmonic -convex functions, we also obtain some new versions of Hermite-Hadamard’s inequality, essentially utilizing the concepts from both ordinary calculus and fractional calculus. In order to show the significance of the main results, we present some applications to special means of real numbers.

Before we proceed further, let us recall some previously known concepts from fractional calculus. The Caputo-Fabrizio fractional derivative and fractional integrals are defined, respectively, as follows.

Let , and then the definition of the left fractional derivative is defined as follows: and the associated fractional integral is defined as follows:

The right Caputo-Fabrizio fractional derivative is defined as follows: and the associated right fractional integral is defined as follows: where is the normalization function satisfying . For more details, see [8, 9].

Our calculations involve beta and hypergeometric functions. For the sake of completeness, let us recall these classical concepts. The beta and hypergeometric functions are defined as or

2. Main Results

In this section, we discuss our main results.

2.1. Uniform Harmonic -Convex Functions

We now introduce the class of uniform harmonic -convex functions.

Definition 4. Let be a real function. A function is said to be a uniform harmonic -convex function, if

Some special cases of Definition 4 are enlisted as follows: (i)If we take , then we obtain the class of uniform harmonic convex function, which is defined as(ii)If we take , then we obtain the class of uniform harmonic -convex function, which is defined as(iii)If we take , then we obtain the class of the Godunova-Levin-Dragomir (GLD) type of uniform harmonic -convex function, which is defined as(iv)If we take and , then we obtain the class of strongly uniform harmonic convex function, which is defined as(v)If we take and , then we obtain the class of approximate harmonic convex function, which is defined as(vi)If we take and , then we obtain the class of approximate harmonic convex function of order , which is defined as

Definition 5. A function is said to be a uniform harmonic convex function, if

Definition 6. A function is said to be a uniform harmonic -convex function, if

Definition 7. A function is said to be a Godunova-Levin-Dragomir (GLD) type of uniform harmonic -convex functions, if

Definition 8. A function is said to be a strongly uniform harmonic convex function, if

Definition 9. A function is said to be an approximate uniform harmonic convex function, if

Definition 10. A function is said to be an approximate uniform harmonic convex function of order , if

2.2. A New Hermite-Hadamard’s Inequality

We now derive a new variant of Hermite-Hadamard’s inequality using the class of uniform harmonic -convex functions. We also discuss some new special cases of this result.

Theorem 11. Let be a uniform harmonic -convex function, and then where .

Proof. Since is a uniform harmonic -convex function, then for , we have Using the change of variable technique, we have After simplifying, we obtain Now, we prove our second inequality using the notion of uniform harmonic convexity. Adding (21) and (22), we have Integrating both sides with respect to on , then we have This completes the proof.

We now discuss some special cases of Theorem 11: (i)If we take in Theorem 11, then we have the result for uniform harmonic convex functions(ii)If we take in Theorem 11, then we have the result for uniform harmonic -convex functions(iii)If we take in Theorem 11, then we have the result for the GLD type of uniform harmonic -convex functions

Corollary 12. Under the assumptions of Theorem 11, if is a uniform harmonic convex function, then

Corollary 13. Under the assumptions of Theorem 11, if is a uniform harmonic -convex function, then

Corollary 14. Under the assumptions of Theorem 11, if is a GLD type of uniform harmonic -convex functions, then

2.3. Fractional Hermite-Hadamard’s Inequality Using Caputo-Fabrizio Fractional Integrals

We now derive a fractional version of Theorem 11 by using Caputo-Fabrizio fractional integrals.

Theorem 15. Let be a uniform harmonic -convex function, and then

Proof. Since is a uniform harmonic -convex function, then The above inequality can be written as Multiplying both sides by , adding , and using Caputo-Fabrizio fractional integrals, we have This implies Thus, we have Now, we compute our second inequality. Since is a uniform harmonic -convex function, we have Multiplying both sides by , adding , and using Caputo-Fabrizio fractional integrals, we have This implies This completes the proof.

Now, we discuss some special cases of Theorem 15: (i)If we take in Theorem 15, then we have the result for uniform harmonic convex functions(ii)If we take and in Theorem 15, then we have the result for strongly harmonic convex functions(iii)If we take in Theorem 15, then we have the result for uniform harmonic -convex functions(iv)If we take and in Theorem 15, then we have the result for strongly harmonic -convex functions(v)If we take in Theorem 15, then we have the result for the GLD type of harmonic -convex functions

Corollary 16. Under the assumptions of Theorem 15, if is a uniform harmonic convex function, then

Corollary 17. Under the assumptions of Theorem 15, if is a strongly harmonic convex function, then

Corollary 18. Under the assumptions of Theorem 15, if is a uniform harmonic -convex function, then

Corollary 19. Under the assumptions of Theorem 15, if is a strongly harmonic -convex function, then

Corollary 20. Under the assumptions of Theorem 15, if is a GLD type of harmonic -convex functions, then

2.4. Dragomir-Agarwal Type of Inequalities

In this section, we derive the Dragomir-Agarwal type of inequalities using the concept of uniform harmonic -convex functions. For this, we first derive a new lemma which will be used as an auxiliary result in obtaining our next results.

Lemma 21. Let be a differentiable function on with . If , , and also , then we have where .

Proof. Let us consider Multiplying both sides by and subtracting , we have This completes the proof.

Theorem 22. If is a uniform harmonic -convex function with and , then we have

Proof. Using Lemma 21, modulus property, and uniform harmonic -convexity of , we have This completes the proof.

Now, we discuss some special cases of Theorem 22: (i)If we take in Theorem 22, then we have the result for uniform harmonic -convex functions(ii)If we take in Theorem 22, then we have the result for uniform harmonic convex functions(iii)If we take in Theorem 22 and by similar proceedings as in the above corollaries, then we have the result for uniform harmonic -convex functions

Corollary 23. Under the assumptions of Theorem 22, if is a uniform harmonic -convex function, then where

Corollary 24. Under the assumptions of Theorem 22, if is a uniform harmonic convex function, then where

Remark 25. In Corollary 24, if we take (1), then we can obtain the result for a strongly harmonic convex function(2), then we can obtain the result for an approximate harmonic convex function(3), then we can obtain the result for an approximate harmonic convex function of order

Corollary 26. Under the assumptions of Theorem 22, if is a uniform harmonic -convex function, then