# Some General Integral Operator Inequalities Associated withφ-Quasiconvex Functions

Authors: Young Chel Kwun; Moquddsa Zahra; Ghulam Farid; Praveen Agarwal; Shin Min Kang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper deals with generalized integral operator inequalities which are established by usingφ-quasiconvex functions. Bounds of an integral operator are established which have connections with different kinds of known fractional integral operators. All the results are deducible for quasiconvex functions. Some fractional integral inequalities are deduced.


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## Body

## 1. Introduction and Preliminaries

Convex functions play a vital role in the theory of mathematical analysis. Many generalizations have been given for the convex function, for example,α-convex, m-convex, h-convex, α,m-convex, h,m-convex, s-convex, s,m-convex, φ-convex, and quasiconvex functions (see [1–10]). We will use φ-quasiconvex functions to study the bounds of unified integral operators, and the established results are directly related with fractional integral operators in particular cases. All the fractional integral operators defined in [11–15] satisfy the results of this paper for φ-quasiconvex functions, and also the results of [16–19] are reproduced in special cases.Definition 1.
(see [20]). A function f:J⟶ℝ is called convex if(1)ftx0+1−ty0≤tfx0+1−tfy0,t∈0,1,holds ∀x0,y0∈J, where J is an interval in ℝ.Definition 2.
(see [21]). A function f:J⟶ℝ is called φ-quasiconvex if(2)ftx0+1−ty0≤maxfy0,fy0+φfx0,fy0,t∈0,1,holds ∀x0,y0∈J, where J is an interval in ℝ and φ:fJ×fJ⟶ℝ is a bifunction.
Forφx0,y0=x0−y0, (2) reduces to quasiconvex function. It is to be noted that every convex function is quasiconvex but converse does not hold.Example 1.
(see [22]). A function f:−2,2⟶ℝ defined as(3)fx=1,x∈−2,−1,x2,x∈−1,2,is quasiconvex on −2,2 but not a convex function on the same interval.
The aim of this paper is to establish integral inequalities by usingφ-quasiconvex functions. The results will provide upper bounds of integral operators for φ-quasiconvex functions, which will behave like compact formulas that unify bounds of various kinds of operators already defined in literature. Next, we give some generalized fractional integral operators connected with the findings of this work.Definition 3.
(see [14]). Let f∈Lx0,y0 and g be positive and increasing function on x0,y0, and also, let g have continuous derivative on x0,y0. The left and right fractional integrals of f with respect to g on x0,y0 of order λ, where λ>0, are given as follows:(4)Igλx0+fx=1Γλ∫x0xgx−gtλ−1g′tftdt,x>x0,Igλy0−fx=1Γλ∫xy0gt−gxλ−1g′tftdt,x<y0,where(5)Γλ=∫0∞tλ−1e−tdt.Definition 4.
(see [23]). Let f∈Lx0,y0 and g be positive and increasing function on x0,y0, and also let g have continuous derivative on x0,y0. The left and right k-fractional integrals of f with respect to g on x0,y0 of order λ, where λ,k>0, are given as follows:(6)Igλx0+kfx=1kΓkλ∫x0xgx−gtλ/k−1g′tftdt,x>x0,Igλy0−kfx=1kΓkλ∫xy0gt−gxλ/k−1g′tftdt,x<y0,where(7)Γkλ=∫0∞tλ−1e−tk/kdt.Definition 5.
(see [24]). Let f∈L1x0,y0 and x∈x0,y0; also, let Φ,λ,ϱ,κ,γ,σ∈ℂ, ℜλ,ℜϱ,ℜκ>0, ℜχ>ℜγ>0 with p≥0, ϑ>0, and 0<k≤ϑ+ℜλ, then the generalized fractional integral operators ϵλ,ϱ,κ,Φ,x0+γ,ϑ,k,χf and ϵλ,ϱ,κ,Φ,y0−γ,ϑ,k,χf are defined as(8)ϵλ,ϱ,κ,Φ,x0+γ,ϑ,k,χfx;p=∫x0xx−tϱ−1Eλ,ϱ,κγ,ϑ,k,χΦx−tλ;pftdt,ϵλ,ϱ,κ,Φ,y0−γ,ϑ,k,χfx;p=∫xy0t−xϱ−1Eλ,ϱ,κγ,ϑ,k,χΦt−xλ;pftdt,where Eλ,ϱ,κγ,ϑ,k,χt;p is given by(9)Eλ,ϱ,κγ,ϑ,k,χt;p=∑n=0∞βpγ+nk,χ−γβγ,χ−γχnkΓλn+ϱtnκnϑ.
In [11], Farid defined a unified integral operator and proved the boundedness, linearity, and continuity of these integrals. It is given in the following definition.Definition 6.
(see [11]). Let f,g:x0,y0⟶ℝ where 0<x0<y0 be the functions such that f is positive and integrable over x0,y0 and g is differentiable and strictly increasing. Also, let Ψ/x be an increasing function on x0,∞ and ϱ,κ,γ,χ∈ℂ, p,λ ,ϑ≥ 0 and 0<k≤ϑ+λ. Then, for x∈x0,y0, the left and right integral operators are defined as(10)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfx,Φ;p=∫x0xGxyEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′yfydy,(11)Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx,Φ;p=∫xy0GyxEλ,,κγ,ϑ,k,χ,g;Ψg′yfydy,where(12)GxyEλ,ϱ,κγ,ϑ,k,χ,g;Ψ=Ψgx−gygx−gyEλ,ϱ,κγ,ϑ,k,χΦgx−gyλ;p.
By making particular choices forΨandg and parameters involved in (9), several fractional integrals can be obtained (see [19], Remarks 6 and 7). In [19], Zhao. et al. proved the bounds of unified integral operators for quasiconvex functions stated in Theorems 1 to 4.Theorem 1.
Considerf:x0,y0⟶ℝ be a positive quasiconvex function and g:x0,y0⟶ℝ be differentiable and strictly increasing function. Also, Ψ/x be an increasing function on x0,y0 and ϱ,κ,γ,χ∈ℂ, p,λ,ν,ϑ≥ 0, 0<k≤ϑ+λ, and 0<k≤ϑ+ν. Then, for x∈x0,y0, we have(13)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfx,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0maxfx0,fx+Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxmaxfx,fy0.Theorem 2.
Under the assumptions of Theorem1, the following result holds:(14)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χy0,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χx0,Φ;p≤Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;p+Eν,ϱ,κγ,ϑ,k,χΦgy0−gx0ν;pmaxfx0,fy0.Theorem 3.
Along with the assumptions of Theorem1, if fx0+y0−x=fx, then the following result holds:(15)fx0+y02Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p+Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p≤Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p+Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p≤Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;p+Eν,ϱ,κγ,ϑ,k,χΦgy0−gx0ν;pmaxfx0,fy0.Theorem 4.
Considerf,g:x0,y0⟶ℝ be two differentiable functions such that f′ is quasiconvex and g be strictly increasing for 0<x0<y0. Also, Ψ/x be an increasing function on x0,y0 and ϱ,κ,γ,χ∈ℂ, p,λ,ν,ϑ≥ 0 and 0<k≤ϑ+λ and 0<k≤ϑ+ν. Then, for x∈x0,y0, we have(16)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0maxf′x0,f′x+Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxmaxf′x,f′y0,where(17)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p=∫x0xGxtEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′tf′tdt,(18)Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p=∫xy0GtxEν,ϱ,κγ,ϑ,k,χ,g;Ψg′tf′tdt.
All of the above results are direct consequences of the results of this paper. Also, some of the results in papers [16–18] are very special cases. In the next section, φ-quasiconvexity has been used frequently to obtain the upper bounds and the Hadamard inequality, which gives upper as well as lower bounds of unified integral operators. Also defining convolution of two functions, some bounds have been obtained for φ-quasiconvexity of f′ of differential function f. In Section 3, some applications of the main results are given. In the whole paper, we use the notation(19)maxfy0,fy0+φfx0,fy0=Mφfx0,y0.


## 2. Main Results

Theorem 5.
Considerf:x0,y0⟶ℝ is φ-quasiconvex and positive and g:x0,y0⟶ℝ differentiable and strictly increasing functions. If Ψ/x is increasing function on x0,y0 and ϱ,κ,γ,χ∈ℂ, p,λ,ν,ϑ≥0, 0<k≤ϑ+λ and 0<k≤ϑ+ν, then for x∈x0,y0, the following inequality holds:(20)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfx,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφfx0,x+Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxMφfx,y0.Proof.
For the kernel defined in (12) and the function g, we can write the following inequality:(21)GxtEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′t≤Gxx0Eλ,ϱ,κγ,ϑ,k,χ,g;Ψg′t,x∈x0,y0,t∈x0,x.
By usingφ-quasiconvexity of f on x0,x, one can get(22)ft≤Mφfx0,x,x∈x0,y0,t∈x0,x.
The following integral inequality is constituted from (21) and (22):(23)∫x0xGxtEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′tftdt≤Mφfx0,xGxx0Eλ,ϱ,κγ,ϑ,k,χ,g;Ψ∫x0xg′tdt.
Using (10) in the left and integrating on the right side of inequality (23), we obtain the following upper bound of the left integral operator:(24)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφfx0,x.
Now, following the similar technique fort∈x,y0 and x∈x0,y0, we can write(25)GtxEν,ϱ,κγ,ϑ,k,χ,g;Ψg′t≤Gy0xEν,ϱ,κγ,ϑ,k,χ,g;Ψg′t.
Usingφ-quasiconvexity for t∈x,y0 and x∈x0,y0, we obtain(26)ft≤Mφfx,y0.
The following integral inequality is constituted from (25) and (26):(27)∫xy0GtxEν,ϱ,κγ,ϑ,k,χ,g;Ψg′tftdt≤Gy0xEν,ϱ,κγ,ϑ,k,χ,g;ΨMφfx,y0∫xy0g′tdt.
Using (11) in the left and integrating on the right side of the above inequality, we obtain the following upper bound of the right integral operator:(28)Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx;p≤Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxMφfx,y0.
By summing (24) and (28), the inequality (20) can be obtained.Corollary 1.
Usingλ=ν in (20), we get the following result:(29)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfx,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφfx0,x+Eλ,ϱ,κγ,ϑ,k,χΦgy0−gxλ;pΨgy0−gxMφfx,y0.Remark 1.
(i)
Forφx0,y0=x0−y0 in (20), we obtain inequality (13) of Theorem 1.(ii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xv/k for the right-hand integral in (20) with p=Φ=0, we obtain Theorem 2.1 in [17].(iii)
Forλ=ν in the resulting inequality of (ii), we obtain Corollary 2.2 in [17].(iv)
Forφx0,y0=x0−y0, Ψx=xλ for the left-hand integral and Ψx=xν for the right-hand integral in (20) with p=Φ=0, we obtain Corollary 2.3 in [17].(v)
Under the same assumptions as in (ii) along withg as identity function, the result (20) reduces to Corollary 2.4 in [17].(vi)
Under the same assumptions as in (iv) along withg as identity function, the result (20) reduces to Corollary 2.5 in [17].(vii)
Under the same assumptions as in (ii), iff is increasing on x0,y0, the result (20) reduces to Corollary 2.6 in [17].(viii)
Under the same assumptions as in (ii), iff is decreasing on x0,y0, the result (20) reduces to Corollary 2.7 in [17].(iix)
Forλ=ν in the resulting inequality of (viii), we obtain Corollary 2.2 in [18].Theorem 6.
The following result holds under the suppositions of Theorem5:(30)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;p+Eν,ϱ,κγ,ϑ,k,χΦgy0−gx0ν;pMφfx0,y0.Proof.
Usingx=y0 in (24) and x=x0 in (28) and then adding the obtained inequalities, we get (30).Corollary 2.
Usingλ=ν in (30), we get the following result:(31)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤2Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;pMφfx0,y0.Remark 2.
(i)
Forφx0,y0=x0−y0 in (30), we obtain inequality (14) of Theorem 2.(ii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xν/k for the right-hand integral in (30) with p=Φ=0, we obtain Theorem 3.1 in [17].(iii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xν/k for the right-hand integral in (31) with p=Φ=0, we obtain Corollary 3.2 in [17].(iv)
Forφx0,y0=x0−y0, replacing Φ with Φ′=Φ/y0−x0λ, Ψx=xλ for the left-hand integral, Ψx=xν for the right-hand integral, and g as identity function in (30), we obtain Theorem 2.1 in [18].(v)
Under the same assumptions as in (iv) along withλ=k=1, the result (30) reduces to Theorem 3.3 in [16].
Before proceeding to the next result, we will prove the following lemma. This lemma is necessary to prove the upcoming result.Lemma 1.
Letf:x0,y0⟶ℝ be φ-quasiconvex function. If fx=fx0+y0−x, then the following inequality holds:(32)fx0+y02≤Mφfx,x,x∈x0,y0.Proof.
Usingφ-quasiconvexity of the function f, the upcoming inequality holds:(33)fx0+y02≤maxfx−x0y0−x0x0+y0−xy0−x0y0,fx−x0y0−x0x0+y0−xy0−x0y0+φfx−x0y0−x0y0+y0−xy0−x0x0,fx−x0y0−x0x0+y0−xy0−x0y0,fx0+y02≤Mφfx,x0+y0−x.
Usingfx0+y0−x=fx in above inequality, we get the required inequality.Remark 3.
Usingφx0,y0=x0−y0, (32) coincides with Lemma 1 in [19].Theorem 7.
Along with the assumptions of Theorem5, if fx0+y0−x=fx and φx,y=x+y, then the following results hold:(34)13fx0+y02Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p≤Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;p+Eν,ϱ,κγ,ϑ,k,χΦgy0−gx0ν;pMφfx0,y0,(35)fx0+y02Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p≤Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p+Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;p+Eν,ϱ,κγ,ϑ,k,χΦgy0−gx0ν;pMφfx0,y0,provided Mφfx,x=fx+φfx,fx or Mφfx,x=fx.Proof.
From the kernel defined in (12) and the function g, we can write(36)Gxx0Eν,ϱ,κγ,ϑ,k,χ,g;Ψg′x≤Gy0x0Eν,ϱ,κγ,ϑ,k,χ,g;Ψg′x,x∈x0,y0.
Usingφ-quasiconvexity of f on x0,y0, we have(37)fx≤Mφfx0,y0,x∈x0,y0.
The following inequality is constituted from (36) and (37):(38)∫x0y0Gxx0Eν,ϱ,κγ,ϑ,k,χ,g;Ψg′xfxdx≤Gy0x0Eν,ϱ,κγ,ϑ,k,χ,g;ΨMφfx0,y0∫x0y0g′xdx.
Using (11) in the left and integrating on the right side of the above inequality, we obtain the following upper bound of the right integral operator:(39)Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤Eν,ϱ,κγ,ϑ,k,χΦgy0−gx0ν;pΨgy0−gx0Mφfx0,y0.
Also,(40)Gy0xEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′x≤Gy0x0Eλ,ϱ,κγ,ϑ,k,χ,g;Ψg′x,x∈x0,y0.
From (37) and (40), we get(41)∫x0y0Gy0xEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′xfxdx≤Gy0x0Eλ,ϱ,κγ,ϑ,k,χ,g;ΨMφfx0,y0∫x0y0g′xdx.
Using (10) in the left and integrating on the right side of the above inequality, we obtain the following upper bound of the left integral operator:(42)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;pΨgy0−gx0Mφfx0,y0.
Now using (32) of Lemma 1, we can have(43)fx0+y02Gxx0Eν,ϱ,κγ,ϑ,k,χ,g;Ψg′x≤Gxx0Eν,ϱ,κγ,ϑ,k,χ,g;Ψg′xMφfx,x.Case 1.
IfMφfx,x=fx+φfx,fx, then by using (11), φx,y=x+y in (43), and integrating over x0,y0, we get(44)fx0+y02Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p≤3Fgν,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p.
In this case, we also have(45)fx0+y02Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p≤3Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p.Case 2.
IfMφfx,x=fx, then by using (11) in (43), we get(46)fx0+y02Fgv,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p≤Fgv,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p.
In this case, we also have(47)fx0+y02Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p≤Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p.
The inequality (34) will be obtained by summing (39) with (42) and (44) with (45) and then combining the resulting inequalities. The inequality (35) will be obtained by summing (39) with (42) and (46) with (47) and then combining the resulting inequalities.Corollary 3.
Forλ=ν in (34) and (35), we get the following results:(48)13fx0+y02Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p≤Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤2Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;pMφfx0,y0,(49)fx0+y02Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χ1y0,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χ1x0,Φ;p≤Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χfy0,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χfx0,Φ;p≤2Ψgy0−gx0Eλ,ϱ,κγ,ϑ,k,χΦgy0−gx0λ;pMφfx0,y0.Remark 4.
(i)
Forφx0,y0=x0−y0 in (34), we get inequality (15) of Theorem 3.(ii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xν/k for the right-hand integral in (34) with p=Φ=0, we obtain Theorem 2.16 in [17].(iii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xν/k for the right-hand integral in (48) with p=Φ=0, we obtain Corollary 2.17 in [17].(iv)
Forφx0,y0=x0−y0, Ψx=xλ for the left-hand integral and Ψx=xν for the right-hand integral in (34) with p=Φ=0, we obtain Corollary 2.18 in [17].(v)
Under the same assumptions as in (ii) along withg as identity function, the result (34) reduces to Corollary 2.19 in [17].(vi)
Under the same assumptions as in (iv) along withg as identity function, the result (34) reduces to Corollary 2.20 in [17].(vii)
Under the same assumptions as in (ii), iff is increasing on x0,y0, the result (34) reduces to Corollary 2.21 in [17].(viii)
Under the same assumptions as in (ii), iff is decreasing on x0,y0, the result (34) reduces to Corollary 2.22 in [17].Theorem 8.
Considerf,g:x0,y0⟶ℝ are two differentiable functions such that f′ is φ-quasiconvex and g is strictly increasing for 0<x0<y0. If Ψ/x is increasing function on x0,y0 and ϱ,κ,γ,χ∈ℂ, p,λ,ν,ϑ≥0, 0<k≤ϑ+λ and 0<k≤ϑ+ν, then for x∈x0,y0, the following inequality holds:(50)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p+Fgv,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφf′x0,x+Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxMφf′x,y0,where Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p and Fgv,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p are defined in (17) and (18).Proof.
Theφ-quasiconvexity of f′ implies the following inequality:(51)f′t≤Mφf′x0,x,t∈x0,x,x∈x0,y0,which is equivalent to(52)−Mφf′x0,x≤f′t≤Mφf′x0,x.
First consider(53)f′t≤Mφf′x0,x.
The following inequality is constituted from (21) and (53):(54)∫x0xGxtEλ,ϱ,κγ,ϑ,k,χ,g;Ψg′tf′tdt≤Mφf′x0,xGxx0Eλ,ϱ,κγ,ϑ,k,χ,g;Ψ∫x0xg′tdt,from which we get(55)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφf′x0,x.
Now, we consider(56)−Mφf′x0,x≤f′t.
Using (21) and (56), we get(57)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p≥−Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφf′x0,x.
Now, again usingφ-quasiconvexity of f′, we have(58)f′t≤Mφf′x,y0,t∈x,y0,x∈x0,y0.
Similarly using (25) and (58), one can obtain(59)Fgv,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p≤Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxMφf′x,y0,(60)Fgv,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p≥−Eν,ϱ,κγ,ϑ,k,χΦgy0−gxν;pΨgy0−gxMφf′x,y0.
The inequality (50) will be obtained by summing (55), (57), (59), and (60).Corollary 4.
Forλ=ν in (50), we get the following result:(61)Fgλ,ϱ,κ,x0+Ψ,γ,ϑ,k,χf∗gx,Φ;p+Fgλ,ϱ,κ,y0−Ψ,γ,ϑ,k,χf∗gx,Φ;p≤Eλ,ϱ,κγ,ϑ,k,χΦgx−gx0λ;pΨgx−gx0Mφf′x0,x+Eλ,ϱ,κγ,ϑ,k,χΦgy0−gxλ;pΨgy0−gxMφf′x,y0.Remark 5.
(i)
Forφx0,y0=x0−y0 in (50), we obtain inequality (16) of Theorem 4.(ii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xν/k for right-hand integral in (50) with p=Φ=0, we obtain Theorem 2.8 in [17].(iii)
Forφx0,y0=x0−y0, Ψx=xλ/k for the left-hand integral and Ψx=xν/k for the right-hand integral in (61) with p=Φ=0, we obtain Corollary 2.9 in [17].(iv)
Forφx0,y0=x0−y0, Ψx=xλ for the left-hand integral and Ψx=xν for the right-hand integral in (50) with p=Φ=0, we obtain Corollary 2.10 in [17].(v)
Under the same assumptions as in (ii) along withg as identity function, the result (50) reduces to Corollary 2.11 in [17].(vi)
Under the same assumptions as in (iv) along withg as identity function, the result (50) reduces to Corollary 2.12 in [17].(vii)
Under the same assumptions as in (ii), iff is increasing on x0,y0, the result (50) reduces to Corollary 2.13 in [17].(viii)
Under the same assumptions as in (ii), iff is decreasing on x0,y0, the result (50) reduces to Corollary 2.14 in [17].(ix)
Under the same assumptions as in (ii), if in addition we putx=x0 and x=y0 in the left- and right-hand integrals, respectively, we obtain Theorem 3.2 in [17].(x)
Forλ=ν in the resulting inequality of (ix), we obtain Corollary 3.5 in [17].(xi)
Forλ=k=1 in the resulting inequality of (x), we obtain Corollary 3.6 in [17].


## 3. Applications

In this section, we present some results by applying theorems of previous section.Proposition 1.
The following result holds under the suppositions of Theorem5:(62)ΓϱIgϱx0+fx+Igϱy0−fx≤gx−gx0ϱMφfx0,x+gy0−gxϱMφfx,y0.Proof.
ForΨt=tϱ,ϱ>0 and p=Φ=0, λ=ν with Ψ/t is increasing for ϱ≥1 in the proof of Theorem 5 we get (62).Proposition 2.
The following result holds under the suppositions of Theorem5:(63)Ix0+Ψfx+Iy0−Ψfx≤Ψx−x0Mφfx0,x+Ψy0−xMφfx,y0.Proof.
Usingg as identity function, Φ=p=0, and λ=ν in the proof of Theorem 5, we get inequality (63).Corollary 5.
ForΨt=Γϱtϱ/k/kΓkϱ in Theorem 5, the following bound for ϱ≥k is satisfied:(64)Igϱx0+kfx+Igϱy0−kfx≤1kΓkϱgx−gx0ϱ/kMφfx0,x+gy0−gxϱ/kMφfx,y0.Corollary 6.
UsingΨt=tϱ in (63), fractional integrals Iϱx0+fx and Iϱy0−fx defined in [14] are obtained which satisfy the following bound:(65)ΓϱIϱx0+fx+Iϱy0−fx≤x−x0ϱMφfx0,x+y0−xϱMφfx,y0.Corollary 7.
UsingΨt=Γϱtϱ/k/kΓkϱ in (63), fractional integral operators Iϱx0+kfx and Iϱy0−kfx given in [26] are obtained which satisfy the following bound:(66)Iϱx0+kfx+Iϱy0−kfx≤1kΓkϱx−x0ϱ/kMφfx0,x+y0−xϱ/kMφfx,y0.Corollary 8.
UsingΨt=tϱ,ϱ>0, and gx=xε/ε,ε>0, in (10) and (11), respectively, with p=Φ=0, then fractional integral operators Iεx0+ϱfx and Iεy0−ϱfx given in [27] are obtained which satisfy the following bound:(67)Iεx0+ϱfx+Iεy0−ϱfx≤1εϱΓϱxε−x0εϱMφfx0,x+y0ε−xεϱMφfx,y0.Corollary 9.
UsingΨt=tϱ,ϱ>0, and gx=xs+1/s+1,s>0, in (10) and (11), respectively, with p=Φ=0, then fractional integral operators Isx0+ϱfx and Isy0−ϱfx are obtained which satisfy the following bound:(68)Isx0+ϱfx+Isy0−ϱfx≤1s+1ϱΓϱxs+1−x0s+1ϱMφfx0,x+y0s+1−xs+1ϱMφfx,y0.Corollary 10.
UsingΨt=Γϱtϱ/k/kΓkϱ and gx=xs+1/s+1,s>0 in (10) and (11), respectively, with p=Φ=0, then fractional integral operators Iksx0+ϱfx and Iksy0−ϱfx given in [28] are obtained which satisfy the following bound:(69)Iksx0+ϱfx+Iksy0−ϱfx≤1s+1ϱ/kkΓkϱxs+1−x0s+1ϱ/kMφfx0,x+y0s+1−xs+1ϱ/kMφfx,y0.Corollary 11.
UsingΨt=tϱ,ϱ>0, gx=xβ+s/β+s,β,s>0, in (10) and (11), respectively, with p=Φ=0, fractional integral operators Iβsx0+ϱfx and Iβsy0−ϱfx are obtained given in [13] which satisfy the following bound:(70)Iβsx0+ϱfx+Iβsy0−ϱfx≤1β+sϱΓϱxβ+s−x0β+sϱMφfx0,x+y0β+s−xβ+sϱMφfx,y0.Corollary 12.
UsingΨt=tϱ,ϱ>0, gx=x−x0ε/ε in (10), and gx=−y0−xε/ε in (11), where ε>0 with p=Φ=0, then following fractional integral operators are obtained given in [12]:(71)Fλ,δ,κ,x0+tϱ,γ,ϑ,κ,χfx=Iεx0+ϱfx=ε1−ϱΓϱ∫x0xx−x0ε−t−x0εϱ−1t−x0ε−1ftdt,Fλ,δ,κ,y0−tϱ,γ,ϑ,κ,χfx=Iεy0−ϱfx=ε1−ϱΓϱ∫xy0y0−xε−y0−tεϱ−1y0−tε−1ftdt.
Furthermore, the following bound is also satisfied:(72)Iεx0+ϱfx+Iεy0−ϱfx≤1εϱΓϱx−x0εϱMφfx0,x+y0−xεϱMφfx,y0.Corollary 13.
ForΨt=Γϱtϱ/k/kΓkϱ,ϱ>k and gx=x−x0ε/ε in (10) and gx=−y0−xε/ε in (11), where ε>0 with p=Φ=0, then the following fractional integral operators are obtained given in [29]:(73)Fλ,δ,κ,x0+Γϱtϱ/k/kΓkϱ,γ,ϑ,κ,χfx=Ikεx0+ϱfx=ε1−ϱ/kkΓkϱ∫x0xx−x0ε−t−x0εϱ/k−1t−x0ε−1ftdt,Fλ,δ,κ,y0−Γϱtϱ/k/kΓkϱ,γ,ϑ,κ,χfx=Ikεy0−ϱfx=ε1−ϱ/kkΓkϱ∫xy0y0−xε−y0−tεϱ/k−1y0−tε−1ftdt.
Furthermore, the following bound is also satisfied:(74)Ikεx0+ϱfx+Ikεy0−ϱfx≤1εϱ/kkΓkϱx−x0εϱ/kMφfx0,x+y0−xεϱ/kMφfx,y0.
Similar bounds can be obtained for Theorems7 and 8 which we leave for the reader.


## 4. Concluding Remarks

A notion namelyφ-quasiconvexity is studied under an integral operator that associates with different kinds of operators independently defined by various authors during the last two decades. The consequences of the results are compiled in the form of corollaries and remarks. Although some of the particular cases are analyzed in Section 3 by applying Theorem 5, the reader can further compute more results as desired by applying other theorems.



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