#### Abstract

Fractional integral inequalities help to solve many difference equations. In this paper, we present some fractional integral inequalities for generalized harmonic nonconvex functions. Moreover, we also present applications of developed inequalities.

#### 1. Introduction

Convex analysis and convex functions have been playing an intensive role in the area of pure as well as applied mathematics [1, 2]. In the last few decades, the study of convex sets and convex functions has gained the attention of many researchers because of its interesting geometric feature and enormous use in economics and optimization theory [3, 4]. Convex analysis is playing a constructive role in economics and optimization theory [5, 6]. The roots of convex analysis are thought provoking and deep which are connected with all the other pure and applied sciences. The classical notion of convexity is not enough to solve modern world problems, so several generalizations and extensions of the notion of convexity are introduced in the literature. The function is convex, iffor all and .

If (1) is strict, then is called strictly convex, and if (1) is reversed, then is said to be concave.

In , Zhang and Wan proposed the notion of a -convex function on , and in , Iscan gave the different definition of a -convex function on . For more details, one can see [7, 912].

In , Noor et al. unified the -convex functions with harmonic convex functions and defined new class of functions, i.e., harmonic nonconvex functions called as harmonic -convexity of functions. For various interesting generalizations of convexity, we refer [10, 1416]. Another interesting generalization of the convex function is convex functions, introduced by Gordji et al. [17, 18].

In the present research, we establish the fractional integral inequalities for generalized harmonic nonconvex functions. Our results generalize many existing inequalities in [13, 19]. This paper is organized as follows. In Section 2, some basic definitions and preliminaries are presented, and our main results are given in Section 3. Section 4 is devoted to the applications of main results. Section 5 draws the conclusion.

#### 2. Basic Definitions

Definition 1. (harmonic convex set) (see ). A set is said to be harmonic convex set, iffor all and .

Definition 2. (harmonic convex function) (see ). A function is said to be harmonic convex function iffor all and .

Definition 3. (-harmonic convex set) (see ). A set is said to be -harmonic convex set, iffor all , , and .

Definition 4. (-harmonic convex function) (see ). A function is called -harmonic convex function iffor all and .

Definition 5. (-convex function) (see ). A function is called -convex function such thatfor all and .

Now we are ready to introduce the definition of generalized harmonic nonconvex function.

Definition 6. (generalized harmonic nonconvex function). A function is called generalized harmonic nonconvex function such thatfor all and .

It can be observed that generalized harmonic nonconvex function reduces to η nonconvex function for and simultaneously.

We observe that by taking in (7), we getfor any and . Also, if we take in (7), we getfor any .

With the help of Lemma 1, we formulated the fractional integral inequalities. For detailed study of fractional integral inequalities, we refer [8, 11, 13, 19, 20].

Lemma 1 (see ). Let be a differentiable function on the interior of . If and , then

#### 3. Main Results

Now, we present our main contribution in convex functions and fractional integral inequalities.

Theorem 1. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on with , then we havewhere

Proof. With the help of Lemma 1 and applying the power mean inequality, we get the following:By the definition of generalized harmonic nonconvex functions, we getwhich is the required result.

Remark 1. If we substitute in Theorem 1, then it reduces to , Theorem 2.2.

For in Theorem 1, we have the following result.

Corollary 1. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with and , then we havewhere the values of are given by (12)–(15), respectively.

Remark 2. If we insert in Corollary 1, we retrace (, Corollary 2.3).

Theorem 2. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere is the upper bound of . Also,

Proof. By the help of Lemma 1 and the inequality of Holder’s integral, we estimated

Using the generalized harmonic nonconvexity of , we obtain the following inequalities:

Hence, combining (22)–(24) yields (19).

Remark 3. If we put in Theorem 2, then it reduces to , Theorem 2.4.

Theorem 3. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvexity on , with , and , then we havewhere

Proof. With the use of Lemma 1 and inequality of Holder’s integral, we estimated

Using the definition of generalized harmonic nonconvex functions, we have

Hence, this completes the result.

Remark 4. If we make the substitution of in Theorem 3, then this reduces to , Theorem 2.5].

#### 4. Applications

This section contains some applications of fractional integral inequalities for generalized harmonic nonconvex functions.

Application 1. For , Theorem 2 reduces to the next result.

Corollary 2. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of and are given in (20) and (21), respectively.

Remark 5. If we insert in Corollary 2, it reduces to , Corollary 3.1.

Application 2. For in Theorem 2, we retrace the next result.

Corollary 3. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of and are given in (20) and (21), respectively.

Remark 6. If we insert in Corollary 3, it reduces to , Corollary 3.2.

Application 3. For in Theorem 2, it yields the following.

Corollary 4. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of and are given in (20) and (21), respectively.

Remark 7. If we insert in Corollary 4, it retraces to , Corollary 3.3.

Application 4. Putting in Theorem 2, we get the following result.

Corollary 5. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of and are given in (20) and (21), respectively.

Remark 8. If we insert in Corollary 5, it traces , Corollary 3.4.

Application 5. For , Theorem 3 captures the following result.

Corollary 6. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of are given by (26)–(29), respectively.

Remark 9. If we insert in Corollary 6, it reduces to , Corollary 3.5.

Application 6. By substituting in Theorem 3, the following result is obtained.

Corollary 7. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of are given by (26)–(29), respectively.

Remark 10. If we insert in Corollary 7, it reduces to , Corollary 3.6].

Application 7. For , Theorem 3 captures the following result.

Corollary 8. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we havewhere the values of are given by (26)–(29), respectively.

Remark 11. If we insert in Corollary 8, it traces , Corollary 3.7.

Application 8. Putting and in Theorem 3, we get the following result.

Corollary 9. Assume a -harmonic convex differentiable function . If is integrable and is a generalized harmonic nonconvex function on , with , and , then we get