Some δ -Tempered Fractional Hermite–Hadamard Inequalities Involving Harmonically Convex Functions and Applications

The main objective of this paper is to obtain some new δ -tempered fractional versions of Hermite–Hadamard’s inequality using the class of harmonic convex functions. In order to show the signiﬁcance of the main results, we also discuss some interesting applications.


Introduction and Preliminaries
A function f: I ⊂ R ⟶ R is said to be convex if (1) In recent years, several new extensions of classical convexity have been proposed in the literature. Iscan [1] introduced the notion of harmonically convex functions as follows.
A function f: I ⊂ (0, ∞) ⟶ R is said to be harmon- Hermite-Hadamard's inequality is one of the most studied results pertaining to convexity property of the functions. is result of Hermite and Hadamard reads as follows.
Let f: I � [] 1 , ] 2 ] ⊂ R ⟶ R be a convex function; then, Iscan [1] extended the classical version of Hermite-Hadamard's inequality using the harmonic convexity property of the function.
Let f: I � [] 1 , ] 2 ] ⊂ (0, ∞) ⟶ R be a harmonically convex function; then, e interrelation between theory of convex functions and theory of integral inequalities has attracted several inequality experts and as result several new versions of classical results have been obtained in the literature. For example, Sarikaya et al. [2] used the concepts of fractional calculus in obtaining the fractional analogue of Hermite-Hadamard's inequality. is idea attracted several researchers and a result number of new refined fractional analogues of classical inequalities have been obtained in the literature. For example, Gurbuz et al. [3] obtained some new refinements of integral inequalities using fractional integral operators of positive real order.İşcan and Wu [4] obtained fractional analogue of Hermite-Hadamard's inequality using the concept of harmonically convex functions Awan et al. [5] obtained conformable fractional Hermite-Hadamard's inequality using the harmonic convexity property of the functions. Iftikhar et al. [6] obtained some local fractional Newton's type inequalities via generalized harmonic convex functions. Recently, Sanli et al. [7] obtained some more new fractional Hermite-Hadamard type of inequalities using the harmonic convexity property of the functions.
In recent years, the classical concepts of fractional calculus have been extended and generalized using novel and innovative ideas. For instance, Meerschaert et al. [8] introduced the concepts where power laws are tempered by an exponential factor and showed that this exponential tempering has both mathematical and practical advantages. is inspired Mohammed et al. [9], and they obtained new generalizations of Hermite-Hadamard's inequality using tempered fractional integrals. Mubeen [10] and Sarikaya and Karaca [11] introduced the notion of δ-fractional integrals. Using the concepts of δ-fractional calculus, Lei et al. [12] obtained Hadamard δ-fractional inequalities of Fejer type using GA-s-convex functions. Luo et al. [13] obtained bounds related to multiparameterized δ-fractional integrals and discussed their applications as well. Awan et al. [14] obtained new fractional analogues of Hermite-Hadamard's inequality using δ-Appell's hypergeometric functions and the harmonic convexity property of the functions. e objective of this paper is also to obtain some new δ-tempered fractional versions of Hermite-Hadamard's inequality using the class of harmonic convex functions. Before we proceed further, let us recall some previously known concepts.
Riemann-Liouville fractional integrals are defined as follows.
en, Riemann-Liouville integrals J α ] 1 where is the well-known Gamma function.
e concept of δ-Riemann-Liouville fractional integral is defined as follows: let F be piecewise continuous on I * � (0, ∞) and integrable on any finite subinterval of I � [0, ∞].
In [8], the authors have described a new variation on the fractional calculus as follows.
with λ ≥ 0 and α > 0. en, right-and left-tempered fractional integrals are defined as We now introduce the δ-tempered fractional integrals.

Journal of Mathematics
If λ � 0, it reduces the incomplete δ-gamma function: Remark 1. For the real numbers α > 0, λ ≥ 1 with δ ≥ 1, we have Proof (1) e proof is straightforward, by using the change of variable technique x � (] 2 − ] 1 )μ. (2) To prove this, we use definition of λ-incomplete δ-gamma function: By changing the order of integration, we obtain Using Remark 1 (1), we have is completes the proof.

Results and Discussion
In this section, we discuss our main results.

A New Version of Hermite-Hadamard's Inequality.
We now derive a new δ-tempered fractional Hermite-Hadamard inequality via harmonically convex function.
Proof. Since f is a harmonic convex function, then is implies Multiplying both sides of the above inequality by τ α− 1 e − (λ(] 2 − ] 1 )τ) δ /δ and integrating with respect to τ on [0, 1], we have Journal of Mathematics 3 is implies us, we have Adding (21) and (22), we have (23) Multiplying the above inequality by is implies (25) is implies Combining (20) and (26), we get the required inequality (15) Journal of Mathematics Now, Similarly,

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Substituting the values of I 1 and I 2 in (28), we get the required result.

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Similarly, Substituting the values of I 1 and I 2 in (32) and using Remark 1 (1), we get the required result.

Theorem 2. Let
Proof. Using Lemma 1, the harmonic convexity of |f ′ | q , and Hölder's inequality, we have is completes our proof.