# Some Fractional Integral Inequalities for a Generalized Class of Nonconvex Functions

Authors: Yeliang Xiao; Muhammad Shoaib Saleem; Faiza Batool
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 ξ:J⟶ℝ is convex, if(1)ξtu+1−tv≤tξu+1−tξv,for all u,v∈J and t∈0,1.If (1) is strict, then ξ is called strictly convex, and if (1) is reversed, then ξ is said to be concave.In [7], Zhang and Wan proposed the notion of a p-convex function on J⊂ℝ, and in [8], Iscan gave the different definition of a p-convex function on J⊂0,∞. For more details, one can see [7, 9–12].In [13], Noor et al. unified the p-convex functions with harmonic convex functions and defined new class of functions, i.e., harmonic nonconvex functions called as harmonic p-convexity of functions. For various interesting generalizations of convexity, we refer [10, 14–16]. Another interesting generalization of the convex function is ϕ 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 [14]). A set J=a1,a2⊆ℝ⟶ℝ is said to be harmonic convex set, if(2)uvtu+1−tv∈J,for all u,v∈J and t∈0,1.Definition 2.
(harmonic convex function) (see [15]). A function ξ:J=a1,a2⊆ℝ⟶ℝ is said to be harmonic convex function if(3)ξuvtu+1−tv≤tξu+1−tξv,for all u,v∈J and t∈0,1.Definition 3.
(p-harmonic convex set) (see [16]). A set J=a1,a2⊆ℝ is said to be p-harmonic convex set, if(4)upvptup+1−tvp1/p∈J,for all u,v∈J, t∈0,1, and p≠0.Definition 4.
(p-harmonic convex function) (see [19]). A function ξ:J⟶ℝ is called p-harmonic convex function if(5)ξupvptup+1−tvp1/p≤1−tξu+tfv,for all u,v∈J and t∈0,1.Definition 5.
(η-convex function) (see [17]). A function ξ:J⟶ℝ is called η-convex function such that(6)ξtu+1−tv≤ξv+tηξu,ξv,for all u,v∈J and t∈0,1.Now we are ready to introduce the definition of generalized harmonic nonconvex function.Definition 6.
(generalized harmonic nonconvex function). A functionξ:J⟶ℝ is called generalized harmonic nonconvex function such that(7)ξupvptup+1−tvp1/p≤ξv+tηξu,ξv,for all u,v∈J and t∈0,1.It can be observed that generalized harmonic nonconvex function reduces toη nonconvex function for p=−1 and ηu,v=u−v simultaneously.We observe that by takingup=vp in (7), we get(8)tηξu,ξv≥0,for any u∈J and t∈0,1. Also, if we take t=1 in (7), we get(9)ηξu,ξv≥0,for any u,v∈J.With the help of Lemma1, we formulated the fractional integral inequalities. For detailed study of fractional integral inequalities, we refer [8, 11, 13, 19, 20].Lemma 1 (see [19]).
Letξ:J=a1,a2⊆ℝ∖0⟶ℝ be a differentiable function on the interior J° of J. If ξ′∈La1,a2 and λ∈0,1, then(10)1−λξ2a1pa2pa1p+a2p1/p+λξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du=a2p−a1p2pa1pa2p∫01/22t−λa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt+∫1/212t−2+λa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt.


## 3. Main Results

Now, we present our main contribution in convex functions and fractional integral inequalities.Theorem 1.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J,q≥1 with σ∈0,1, then we have(11)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2pk1p,a1,a21−1/qk1p,a1,a2ξ′a2q+k3p,a1,a2ηξ′a1q,ξ′a2q1/q+k2p,a2,a11−1/qk2p,a2,a1ξ′a2q+k4p,a2,a1ηξ′a1q,ξ′a2q1/q,where(12)k1p,a1,a2=∫01/22t−σa1pa2pta1p+1−ta2p1+1/pdt,(13)k2p,a2,a1=∫1/212t−t+σra1pa2pta1p+1−ta2p1+1/pdt,(14)k3p,a1,a2=∫01/2t2t−σa1pa2pta1p+1−ta2p1+1/pdt,(15)k4p,a2,a1=∫1/21t2t−t+σa1pa2pta1p+1−ta2p1+1/pdt.Proof.
With the help of Lemma1 and applying the power mean inequality, we get the following:(16)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p∫01/22t−σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt+∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt≤a2p−a1p2pa1pa2p∫01/22t−σa1pa2pta1p+1−ta2p1+1pdt1−1q∫01/22t−σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pqdt1/q+∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pdt1−1/p∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pqdt1/q.
By the definition of generalized harmonic nonconvex functions, we get(17)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p∫01/22t−σa1pa2pta1p+1−ta2p1+1/pdt1−1/q∫01/22t−σa1pa2pta1p+1−ta2p1+1/pξ′a2q+tηξ′a1q,ξ′a2qdt1/q+∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pdt1−1/q∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pξ′a2q+tηξ′a1q,ξ′a2qdt1/q=a2p−a1p2pa1pa2pk1p,a1,a21−1/qk1p,a1,a2ξ′a2q+k3p,a1,a2ηξ′a1q,ξ′a2q1/q+k2p,a2,a11−1/qk2p,a2,a1ξ′a2q+k4p,a2,a1ηξ′a1q,ξ′a2q1/q,which is the required result.Remark 1.
If we substituteηu,v=u−v in Theorem 1, then it reduces to [19], Theorem 2.2.Forq=1 in Theorem 1, we have the following result.Corollary 1.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with q=1 and σ∈0,1, then we have(18)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p×k1p,a1,a2ξ′a2+k3p,a1,a2ηξ′a1,ξ′a2+k2p,a2,a1ξ′a2+k4p,a2,a1ηξ′a1,ξ′a,where the values of k1,k2,k3,k4 are given by (12)–(15), respectively.Remark 2.
If we insertηu,v=u−v in Corollary 1, we retrace ([19], Corollary 2.3).Theorem 2.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(19)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2pk5r,p;a1,a2ξ′a1q+ξ′2a1pa2p/a1p+a2p1/pq2+Mη21/q+k6r,p;a2,a1ξ′2a1pa2p/a1p+a2p1/pq+ξ′a2q2+Mη21/q,where Mη is the upper bound of η. Also,(20)k5r,p;a1,a2=∫01/22t−σra1pa2pξa1p+1−ta2pr+r/pdt1/r,(21)k6r,p;a2,a1=∫1/212t−2+σra1pa2pta1p+1−ta2pr+r/pdt1/r.Proof. By the help of Lemma1 and the inequality of Holder’s integral, we estimated(22)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p∫01/22t−σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt+∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt≤a2p−a1p2pa1pa2p∫01/22t−σa1pa2pta1p+1−ta2p1+1/prdt1/rξ′a1pa2pta1p+1−ta2p1/pqdt1/q+∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/prdt1/rξ′a1pa2pta1p+1−ta2p1/pqdt1/q≤a2p−a1p2pa1pa2p∫01/22t−σra1pa2pta1p+1−ta2pr+1/pdt1/rpa1pa2pa2p−a1p∫a2a1pa2p/a1p+a2p1/pξ′uqup+1du1/q+∫1/212t−2+σra1pa2pta1p+1−ta2pr+1/pdt1/rpa1pa2pa2p−a1p∫2a1pa2p/a1p+a2p1pa2ξ′uqup+1du1/q.Using the generalized harmonic nonconvexity ofξ′q, we obtain the following inequalities:(23)pa1pa2pa2p−a1p∫a12a1pa2p/a1p+a2p1/pξ′uqup+1du≤12ξ′a1q+ξ′2a1pa2pa1p+a2p1/p+Mη2,(24)pa1pa2pa2p−a1p∫2a1pa2p/a1p+a2p1pa2ξ′uqup+1du≤12ξ′2a1pa2pa1p+a2p1/p+ξ′a2q+Mη2.Hence, combining (22)–(24) yields (19).Remark 3.
If we putηu,v=u−v in Theorem 2, then it reduces to [19], Theorem 2.4.Theorem 3.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and |ξ′|q is a generalized harmonic nonconvexity on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(25)1−σξ2a1pa2pa1p+a2p1p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p×σr+1+1−σr+12r+11/rk7q,p;a1,a2|ξ′a2|q+k9q,p;a1,a2ηξ′a1q,ξ′a2q1/q+k10q,p;a2,a1ξ′a2q+k8q,p;a2,a1ηξ′a1q,ξ′a2q1/q,where(26)k7q,p;a1,a2=∫01/2a1pa2pta1p+1−ta2pq+q/pdt,(27)k8q,p;a2,a1=∫1/21ta1pa2pta1p+1−ta2pq+q/pdt,(28)k9q,p;a1,a2=∫01/2ta1pa2pta1p+1−ta2pq+q/pdt,(29)k10q,p;a2,a1=∫1/21a1pa2pta1p+1−ta2pq+q/pdt.Proof. With the use of Lemma1 and inequality of Holder’s integral, we estimated(30)1−σξ2a1pa2pa1p+a2p1/p+σξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p∫01/22t−σa2pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt+∫1/212t−2+σa1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pdt≤a2p−a1p2pa1pa2p∫01/22t−σr1/ra1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1/pqdt1/q+∫1/212t−2+σr1/ra1pa2pta1p+1−ta2p1+1/pξ′a1pa2pta1p+1−ta2p1pqdt1/q≤a2p−a1p2pa1pa2p∫01/22t−σr1/ra1pa2pta1p+1−ta2pq+q/pξ′a1pa2pta1p+1−ta2p1/pqdt1/q+∫1/212t−2+σr1/ra1pa2pta1p+1−ta2pq+q/pξ′a1pa2pta1p+1−ta2p1/pqdt1/q.Using the definition of generalized harmonic nonconvex functions, we have(31)≤a2p−a1p2pa1pa2p∫01/22t−σrdt1/r×a1pa2pta1p+1−ta2pq+q/pξ′a2q+tηξ′a1q,ξ′a2q1/q+∫1/212t−2+σrdt1/ra1pa2pta1p+1−ta2pq+q/pξ′a2q+tηξ′a1q,ξ′a2q1/q.Hence, this completes the result.Remark 4.
If we make the substitution ofηu,v=u−v in Theorem 3, then this reduces to [19], Theorem 2.5].


## 4. Applications

This section contains some applications of fractional integral inequalities for generalized harmonic nonconvex functions.Application 1.
Forσ=0, Theorem 2 reduces to the next result.Corollary 2.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0\⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(32)ξ2a1pa2pa1p+a2p1/p−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2pk5r,p;a1,a2ξ′a1q+ξ′2a1pa2p/a1p+a2p1/pq2+Mη21/q+k6r,p;a2,a1ξ′2a1pa2p/a1p+a2p1/pq+ξ′a2q2+Mη21/q,where the values of k5 and k6 are given in (20) and (21), respectively.Remark 5.
If we insertηu,v=u−v in Corollary 2, it reduces to [19], Corollary 3.1.Application 2.
Forσ=1 in Theorem 2, we retrace the next result.Corollary 3.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(33)ξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2pk5r,p;a1,a2ξ′a1q+ξ′2a1pa2p/a1p+a2p1/pq2+Mη21/q+k6r,p;a2,a1ξ′2a1pa2p/a1p+a2p1/pq+ξ′a2q2+Mη21/q,where the values of k5 and k6 are given in (20) and (21), respectively.Remark 6.
If we insertηu,v=u−v in Corollary 3, it reduces to [19], Corollary 3.2.Application 3.
Forσ=1/3 in Theorem 2, it yields the following.Corollary 4.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(34)16ξa1+4ξ2a1pa2pa1p+a2p1/p+ξa2−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2pk5r,p;a1,a2ξ′a1q+ξ′2a1pa2p/a1p+a2p1/pq2+Mη21/q+k6r,p;a2,a1ξ′2a1pa2p/a1p+a2p1/pq+ξ′a2q2+Mη21/q,where the values of k5 and k6 are given in (20) and (21), respectively.Remark 7.
If we insertηu,v=u−v in Corollary 4, it retraces to [19], Corollary 3.3.Application 4.
Puttingσ=1/2 in Theorem 2, we get the following result.Corollary 5.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(35)14ξa1+2ξ2a1pa2pa1p+a2p1/p+ξa2−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2pk5r,p;a1,a2ξ′a1q+ξ′2a1pa2p/a1p+a2p1/pq2+Mη21/q+k6r,p;a2,a1ξ′2a1pa2p/a1p+a2p1/pq+ξ′a2q2+Mη21/q,where the values of k5 and k6 are given in (20) and (21), respectively.Remark 8.
If we insertηu,v=u−v in Corollary 5, it traces [19], Corollary 3.4.Application 5.
Forσ=0, Theorem 3 captures the following result.Corollary 6.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(36)ξ2a1pa2pa1p+a2p1/p+−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p×σr+1+1−σr+12r+11/rk7q,p;a1,a2ξ′a2q+k9q,p;a1,a2ηξ′a1q,ξ′a2q1/q+k10q,p;a2,a1ξ′a2q+k8q,p;a2,a1ηξ′a1q,ξ′a2q1/q,where the values of k7,k8,k9,k10 are given by (26)–(29), respectively.Remark 9.
If we insertηu,v=u−v in Corollary 6, it reduces to [19], Corollary 3.5.Application 6.
By substitutingσ=1 in Theorem 3, the following result is obtained.Corollary 7.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(37)ξa1+ξa22−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p×σr+1+1−σr+12r+11/rk7q,p;a1,a2ξ′a2q+k9q,p;a1,a2ηξ′a1q,ξ′a2q1/q+k10q,p;a2,a1ξ′a2q+k8q,p;a2,a1ηξ′a1q,ξ′a2q1/q,where the values of k7,k8,k9,k10 are given by (26)–(29), respectively.Remark 10.
If we insertηu,v=u−v in Corollary 7, it reduces to [19], Corollary 3.6].Application 7.
Forσ=1/3, Theorem 3 captures the following result.Corollary 8.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we have(38)164ξ2a1pa2pa1p+a2p1/p+ξa1+ξa2−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p×σr+1+1−σr+12r+11/rk7q,p;a1,a2ξ′a2q+k9q,p;a1,a2ηξ′a1q,ξ′a2q1/q+k10q,p;a2,a1ξ′a2q+k8q,p;a2,a1ηξ′a1q,ξ′a2q1/q,where the values of k7,k8,k9,k10 are given by (26)–(29), respectively.Remark 11.
If we insertηu,v=u−v in Corollary 8, it traces [19], Corollary 3.7.Application 8.
Puttingσ=1/2 and ηu,v=u−v in Theorem 3, we get the following result.Corollary 9.
Assume ap-harmonic convex differentiable function ξ:J=a1,a2⊆ℝ∖0⟶ℝ. If ξ′ is integrable and ξ′q is a generalized harmonic nonconvex function on J, with r,q≥1, 1/r+1/q=1 and σ∈0,1, then we get(39)142ξ2a1pa2pa1p+a2p1/p+ξa1+ξa2−pa1pa2pa2p−a1p∫a1a2ξuup+1du≤a2p−a1p2pa1pa2p×σr+1+1−σr+12r+11/rk7q,p;a1,a2ξ′a2q+k9q,p;a1,a2ηξ′a1q,ξ′a2q1/q+k10q,p;a2,a1ξ′a2q+k8q,p;a2,a1ηξ′a1q,ξ′a2q1/q,where the values of k7,k8,k9,k10 are given by (26)–(29), respectively.Remark 12.
If we insertηu,v=u−v in Corollary 9, it reduces to [19], Corollary 3.8.


## 5. Concluding Remarks

Convex functions cannot solve nonconvex problems, and many problems can be encoded as nonconvex optimization problems. In this paper, we investigated a class of generalized harmonic nonconvex functions. We established some fractional integral inequalities for generalized harmonic nonconvex functions. Moreover, we also presented applications of developed inequalities. Our results extend and generalize many existing results in the literature.



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