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

Abstract

The main objective of this paper is to obtain some new -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.

1. Introduction and Preliminaries

A function is said to be convex if

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 is said to be harmonically convex if

Hermite–Hadamard’s inequality is one of the most studied results pertaining to convexity property of the functions. This result of Hermite and Hadamard reads as follows.

Let 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 be a harmonically convex function; then,

The 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. This 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. This 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 -fractional integrals. Using the concepts of -fractional calculus, Lei et al. [12] obtained Hadamard -fractional inequalities of Fejer type using --convex functions. Luo et al. [13] obtained bounds related to multiparameterized -fractional integrals and discussed their applications as well. Awan et al. [14] obtained new fractional analogues of Hermite–Hadamard’s inequality using -Appell’s hypergeometric functions and the harmonic convexity property of the functions.

The objective of this paper is also to obtain some new -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.

Definition 1. (see [15]). Let . Then, Riemann–Liouville integrals and of order with are defined bywhereis the well-known Gamma function.

The concept of -Riemann–Liouville fractional integral is defined as follows: let be piecewise continuous on and integrable on any finite subinterval of . Then, for , we consider -Riemann–Liouville fractional integral of of order

If , then -Riemann–Liouville fractional integrals reduce to classical Riemann–Liouville fractional integral.

In [8], the authors have described a new variation on the fractional calculus as follows.

Definition 2. Let with and . Then, right- and left-tempered fractional integrals are defined as

We now introduce the -tempered fractional integrals.

Definition 3. Let with and . Then, right- and left-tempered -fractional integrals are defined as

Definition 4. For the real numbers with , we define the -incomplete -gamma function byIf , it reduces the incomplete -gamma function:

Remark 1. For the real numbers with , we have(1)(2)

Proof. (1)The proof is straightforward, by using the change of variable technique .(2)To prove this, we use definition of -incomplete -gamma function:By changing the order of integration, we obtainUsing Remark 1 (1), we haveThis completes the proof.

2. Results and Discussion

In this section, we discuss our main results.

2.1. A New Version of Hermite–Hadamard’s Inequality

We now derive a new -tempered fractional Hermite–Hadamard inequality via harmonically convex function.

Theorem 1. Let be an harmonically convex function on with ; then,where and for all .

Proof. Since is a harmonic convex function, thenThis impliesMultiplying both sides of the above inequality by and integrating with respect to on , we haveThis impliesThus, we haveAlso,Adding (21) and (22), we haveMultiplying the above inequality by and integrating with respect to on , we haveThis impliesThis impliesCombining (20) and (26), we get the required inequality (15)

Remark 2. (1)For and in (15), we obtain the classical Hermite–Hadamard’s type inequality for the notion of harmonic convexity(2)For and in 2.1, we obtain the fractional version of Hermite–Hadamard’s type inequality for the notion of harmonic convexity

2.2. New Auxiliary Results

We now derive two new auxiliary results. These results will play a significant role in the development of our next results.

Lemma 1. Let be function; then,for all .

Proof. ConsiderNow,Similarly,Substituting the values of and in (28), we get the required result.

Lemma 2. Let be function; then,where .

Proof. ConsiderUsing Remark 1 (1), we haveSimilarly,Substituting the values of and in (32) and using Remark 1 (1), we get the required result.

2.3. Some More New Hermite–Hadamard’s Type of Integral Inequalities

Now using auxiliary results obtained in the previous section, we derive some new Hermite–Hadamard’s type of integral inequalities.

Theorem 2. Let be function. If , is harmonically convex on with , then

Proof. Using Lemma 1, the harmonic convexity of , and Hölder’s inequality, we haveThis completes our proof.

Theorem 3. Let be an harmonically convex function on with ; then, we havefor all .

Proof. Since is an harmonically convex function, then we haveThis impliesMultiplying both sides of the above inequality by and integrating with respect to on , we haveThus, we haveUsing Remark 1 (1), we haveAlso,Multiplying the above inequality by and integrating with respect to on , we haveUsing Remark 1 (1), we haveThus, we haveCombining (42) and (46), we get the required inequality (37).

Theorem 4. If is function, using the harmonic convexity property of , on with , we have

Proof. Using Lemma 2 and the harmonic convexity property of , we haveChanging the order of integration, we haveThis completes our proof.

Theorem 5. Let be function. If , is harmonically convex on with , then we havewhere

Proof. Using Lemma 2, the harmonic convexity property of , and power mean inequality, we haveChanging the order of integration and after simple calculation, we get the required result.

Corollary 1. All the assumption of Theorem 2 are true; for and , we have

3. Applications

In this section, we investigate some applications to special means and special functions.

3.1. Applications to Special Means

We now recall some special means for two nonnegative numbers.(1)The arithmetic mean: (2)The geometric mean: (3)The harmonic mean: (4)The logarithmic mean: (5)The -logarithmic mean:

Proposition 1. Let ; then, we have

Proof. The proof is simply obtained by using in Corollary 1.

Proposition 2. Let ; then, we have

Proof. The proof is straightforward, by using in Corollary 1.

3.2. Bessel Functions

Consider the function , defined by , and we recall modified Bessel functions of first kind and its first derivative formula, respectively:

Proposition 3. For with , then we have

For a particular case, we have

Proof. The proof is obtained by taking and using it in Corollary 1. For the second inequality, we use relation and .

3.3. -Digamma Functions

Let . Then, -digamma function is defined as

For and , the -digamma function is defined as

Proposition 4. For , such that ,

Proof. The proof is straightforward, by considering and using Remark 2.

Proposition 5. For , such that ,

Proof. The proof is straightforward, by considering and using Corollary 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by HEC Pakistan under project 8081/Punjab/NRPU/R&D/HEC/2017.