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

In this paper, we introduce the notion of uniform harmonic χ-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 χ-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.


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 associ-ated 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.
holds for all x, y ∈ H and i ∈ ½0, 1.
He also obtained a new variant of Hermite-Hadamard's inequality pertaining to harmonic convex functions as follows.
In [7], the class of uniformly convex functions was defined as follows.
Definition 3. Let K ⊂ ℝ be a convex set. A function Y : K ⟶ ℝ is said to be uniformly convex with modulus ψ : ½0, ∞ ⟶ ½0,∞ if ψ is increasing, ψ vanishes only at 0, and The main motivation of this paper is to introduce the class of uniform harmonic χ-convex functions. We show that this class contains some other new classes of uniform harmonic convex functions. Using the class of uniform harmonic χ-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 Y ∈ H 1 ð♭ 1 , ♭ 2 Þ, ♭ 1 < ♭ 2 , α ∈ ½0, 1, 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 BðαÞ > 0 is the normalization function satisfying B ð0Þ = Bð1Þ = 1. 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

Main Results
In this section, we discuss our main results.

Uniform Harmonic χ-Convex
Functions. We now introduce the class of uniform harmonic χ-convex functions.
Definition 4. Let χ : ð0, 1Þ ⟶ ℝ be a real function. A function Y : ½♭ 1 , ♭ 2 ⊂ ð0,∞Þ ⟶ ℝ is said to be a uniform harmonic χ-convex function, if Some special cases of Definition 4 are enlisted as follows: (i) If we take χðiÞ = i, then we obtain the class of uniform harmonic convex function, which is defined as (ii) If we take χðiÞ = i s , then we obtain the class of uniform harmonic s-convex function, which is defined as (iii) If we take χðiÞ = i −s , then we obtain the class of the Godunova-Levin-Dragomir (GLD) type of uniform harmonic s-convex function, which is defined as (iv) If we take χðiÞ = i and ψðiÞ = δi 2 , then we obtain the class of strongly uniform harmonic convex function, which is defined as 2 Journal of Function Spaces (v) If we take χðiÞ = i and ψðiÞ = −δ|i|, then we obtain the class of approximate harmonic convex function, which is defined as (vi) If we take χðiÞ = i and ψðiÞ = −δjij p , then we obtain the class of approximate harmonic convex function of order p > 0, which is defined as Definition 8. A function Y : ½♭ 1 , ♭ 2 ⊂ ð0,∞Þ ⟶ ℝ is said to be a strongly uniform harmonic convex function, if Definition 9. A function Y : ½♭ 1 , ♭ 2 ⊂ ð0,∞Þ ⟶ ℝ is said to be an approximate uniform harmonic convex function, if Definition 10. A function Y : ½♭ 1 , ♭ 2 ⊂ ð0,∞Þ ⟶ ℝ is said to be an approximate uniform harmonic convex function of order p > 0, 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 χ-convex functions. We also discuss some new special cases of this result.
Theorem 11. Let Y : ½♭ 1 , ♭ 2 ⊂ ð0,∞Þ ⟶ ℝ be a uniform harmonic χ-convex function, and then where Proof. Since Y is a uniform harmonic χ-convex function, Using the change of variable technique, we have After simplifying, we obtain Now, we prove our second inequality using the notion of uniform harmonic χ convexity.

Journal of Function Spaces
Adding (21) and (22), we have Integrating both sides with respect to i on ½0, 1, then we have This completes the proof.
We now discuss some special cases of Theorem 11: (i) If we take χðiÞ = i in Theorem 11, then we have the result for uniform harmonic convex functions Corollary 13. Under the assumptions of Theorem 11, if Y is a uniform harmonic s-convex function, then Corollary 14. Under the assumptions of Theorem 11, if Y is a GLD type of uniform harmonic s-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.
Corollary 17. Under the assumptions of Theorem 15, if Y is a strongly harmonic convex function, then Corollary 20. Under the assumptions of Theorem 15, if Y is a GLD type of harmonic s-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 χ-convex functions. For this, we first derive a new lemma which will be used as an auxiliary result in obtaining our next results.
Proof. Using Lemma 21, modulus property, and uniform harmonic χ-convexity of jY ′j, we have This completes the proof. where where where Proof. Using Lemma 21, modulus property, and uniform harmonic χ-convexity of jY ′ j q , then we have Remark 33. In Corollary 32, if we take (1) ψ = δi 2 , then we can obtain the result for a strongly harmonic s-convex function (2) ψ = −δ | i|, then we can obtain the result for an approximate harmonic s-convex function (3) ψ = −δjij p , then we can obtain the result for an approximate harmonic s-convex function of order p > 0 Theorem 34. If Y : ½♭ 1 , ♭ 2 ⟶ ℝ is a uniform harmonic χ-convex function with ♭ 1 < ♭ 2 , 1/p + 1/q = 1, and i ∈ ½0, 1, then