In this paper, we at first develop a generalized integral identity by associating Riemann-Liouville (RL) fractional integral of a function concerning another function. By using this identity estimates for various convexities are accomplish which are fractional integral inequalities. From our results, we obtained bounds of known fractional results which are discussed in detail. As applications of the derived results, we obtain the mid-point-type inequalities. These outcomes might be helpful in the investigation of the uniqueness of partial differential equations and fractional boundary value problems.
Citation: Dumitru Baleanu, Muhammad Samraiz, Zahida Perveen, Sajid Iqbal, Kottakkaran Sooppy Nisar, Gauhar Rahman. Hermite-Hadamard-Fejer type inequalities via fractional integral of a function concerning another function[J]. AIMS Mathematics, 2021, 6(5): 4280-4295. doi: 10.3934/math.2021253
[1] | Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak . Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546 |
[2] | Hasan Kara, Hüseyin Budak, Mehmet Eyüp Kiriş . On Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions. AIMS Mathematics, 2020, 5(5): 4681-4701. doi: 10.3934/math.2020300 |
[3] | Yanping Yang, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function. AIMS Mathematics, 2021, 6(11): 12260-12278. doi: 10.3934/math.2021710 |
[4] | Sabila Ali, Shahid Mubeen, Rana Safdar Ali, Gauhar Rahman, Ahmed Morsy, Kottakkaran Sooppy Nisar, Sunil Dutt Purohit, M. Zakarya . Dynamical significance of generalized fractional integral inequalities via convexity. AIMS Mathematics, 2021, 6(9): 9705-9730. doi: 10.3934/math.2021565 |
[5] | Thabet Abdeljawad, Muhammad Aamir Ali, Pshtiwan Othman Mohammed, Artion Kashuri . On inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional integrals. AIMS Mathematics, 2021, 6(1): 712-725. doi: 10.3934/math.2021043 |
[6] | Ghulam Farid, Hafsa Yasmeen, Hijaz Ahmad, Chahn Yong Jung . Riemann-Liouville Fractional integral operators with respect to increasing functions and strongly $ (\alpha, m) $-convex functions. AIMS Mathematics, 2021, 6(10): 11403-11424. doi: 10.3934/math.2021661 |
[7] | Maimoona Karim, Aliya Fahmi, Shahid Qaisar, Zafar Ullah, Ather Qayyum . New developments in fractional integral inequalities via convexity with applications. AIMS Mathematics, 2023, 8(7): 15950-15968. doi: 10.3934/math.2023814 |
[8] | Shuang-Shuang Zhou, Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu . New Hermite-Hadamard type inequalities for exponentially convex functions and applications. AIMS Mathematics, 2020, 5(6): 6874-6901. doi: 10.3934/math.2020441 |
[9] | Jamshed Nasir, Shahid Qaisar, Saad Ihsan Butt, Hassen Aydi, Manuel De la Sen . Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications. AIMS Mathematics, 2022, 7(3): 3418-3439. doi: 10.3934/math.2022190 |
[10] | XuRan Hai, ShuHong Wang . Hermite-Hadamard type inequalities based on the Erdélyi-Kober fractional integrals. AIMS Mathematics, 2021, 6(10): 11494-11507. doi: 10.3934/math.2021666 |
In this paper, we at first develop a generalized integral identity by associating Riemann-Liouville (RL) fractional integral of a function concerning another function. By using this identity estimates for various convexities are accomplish which are fractional integral inequalities. From our results, we obtained bounds of known fractional results which are discussed in detail. As applications of the derived results, we obtain the mid-point-type inequalities. These outcomes might be helpful in the investigation of the uniqueness of partial differential equations and fractional boundary value problems.
Fractional calculus is an important research field, not only in pure mathematics but in applied mathematics, physics, biology, engineering, economics, etc., as well. In fact, by considering derivatives and integrals of arbitrary real or complex order, we may model more efficiently certain real phenomena. Applications have been found e.g., in human body modelling [1,2], heat conduction [3], viscoelasticity [4,5], time series analysis [6,7], circuits [8], material sciences [9], shear waves [10], etc. Fractional integral operators play a leading and keen role in the development of fractional calculus. The first formulation of a fractional integral operator is due to a continuous study of well-renowned mathematicians and physicists. This fractional integral is well known as RL fractional integral operator. After its existence, many other fractional integral and fractional derivative operators have been introduced. Recently, scientists in their diverse fields are working in the environment of fractional calculus in new directions of respective fields developing rapidly.
Convex functions are very close to the theory of inequalities. Many known and useful inequalities are consequences of convex functions. Some very natural inequalities for example Jensen inequality, Hadamard inequality interpret convex functions beautifully. Fractional integral inequalities are very useful in the study of fractional partial as well as ordinary differential equations. These inequalities establish the uniqueness and bounds of their solutions. In this paper, we study a general form of RL fractional integrals via convex functions. We start with the definition of a convex function.
In 1905 Jensen [11] present the definition of convex function as follows.
Definition 1.1. A function ψ:[a,b]→R is said to be convex if the inequality
ψ(θx+(1−θ)y)≤θψ(x)+(1−θ)ψ(y) |
holds for all x,y∈[a,b] and all θ∈[0,1].
Brechner introduce the definition of s-convex function in [12] which is defined as follows.
Definition 1.2. A function ψ:(0,∞)→R is said to be s-convex in the second sense if the inequality
ψ(θx+ζy)≤θsψ(x)+ζsψ(y) |
holds for some fixed s∈(0,1] and for all x,y∈(0,∞),,θ,ζ>0, where θ+ζ=1. The class of s-convex functions of the second sense usually denoted by K2s.
Definition 1.3. The incomplete beta function Bx(θ,ζ) [13] is defined by
Bx(θ,ζ)=x∫0yθ−1(1−y)ζ−1dy, |
where 0<x<1, θ,ζ>0.
Definition 1.4. The left-sided and right-sided RL fractional integrals Iϱθ+ψ and Iϱζ−ψ of order ϱ>0 on a finite interval [θ,ζ] are defined by
Iϱθ+ψ(x)=1Γ(ϱ)x∫θ(x−ν)ϱ−1ψ(ν)dν,x>θ, |
and
Iϱζ−ψ(x)=1Γ(ϱ)ζ∫x(ν−x)ϱ−1ψ(ν)dν, x<ζ |
respectively. Here, Γ(.) represents the usual Euler gamma function with integral representation
Γ(x)=∞∫0νx−1e−νdν,Re(x)>0. |
Now, we recall the definition of fractional integrals of real valued function concerning to another function [14].
Definition 1.5. Let p:[θ,ζ]→R be an increasing and positive function on (θ,ζ], having a continuous derivative p′ on (θ,ζ). The left and right-sided RL fractional integrals of ψ with respect to the function p on [θ,ζ] of order ϱ>0 are defined respectively by
Iϱθ+;pψ(x)=1Γ(ϱ)x∫θp′(ν)[p(x)−p(ν)]1−ϱψ(ν)dν,x>θ |
Iϱζ−;pψ(x)=1Γ(ϱ)ζ∫xp′(ν)[p(ν)−p(x)]1−ϱψ(ν)dν,x<ζ |
provided that the integrals exists.
Note that
(i) If we take p(x)=x, then we get the Definition 1.4 of RL fractional integrals.
(ii) If we take p(x)=xvv, v>0, then we get the definition of Katugampola fractional integrals given in [15].
(iii) If we take p(x)=xν+sν+s, then we get the definition of generalized conformable fractional integrals defined by Khan et al. in [16].
(iv) If we take p(x)=(x−θ)ss, s>0 in left and p(x)=(ζ−x)ss, s>0 in right, then we get the definition of conformable fractional integrals defined by Jarad et al. in [17].
In this section, we develop an integral identity in the form of the left-sided and right-sided RL fractional integrals of a function concerning to another function. With the help of this identity, we further establish some new inequalities for classical convex and s-convex function.
Lemma 2.1. Let p:[θ,ζ]→R be an increasing and positive function on (θ,ζ], having a continuous derivative p′ on (θ,ζ). Let ψ:[θ,ζ] be a differentiable mapping on (θ,ζ) with θ<ζ and g:[θ,ζ]→R be bounded. If ψ′,g∈L[θ,ζ], then following identity holds:
ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]=1Γ(ϱ)ζ∫θκ(ν)ψ′(ν)dν, |
where
κ(t)={t∫θp′(u)[p(u)−p(θ)]1−ϱg(u)du, t∈[θ,θ+ζ2],t∫ζp′(u)[p(ζ)−p(u)]1−ϱg(u)du, t∈[θ+ζ2,ζ]. |
Proof. Consider
I=ζ∫θκ(ν)ψ′(ν)dν=θ+ζ2∫θκ(ν)ψ′(ν)dν+ζ∫θ+ζ2κ(ν)ψ′(ν)dν=I1+I2. | (2.1) |
Now, integrating I1 by parts, we get
I1=θ+ζ2∫θκ(ν)ψ′(ν)dν=κ(θ+ζ2)ψ(θ+ζ2)−θ+ζ2∫θp′(ν)[p(ν)−p(θ)]1−ϱg(ν)ψ(ν)dν=ψ(θ+ζ2)θ+ζ2∫θp′(u)[p(u)−p(θ)]1−ϱg(u)du−θ+ζ2∫θp′(ν)[p(ν)−p(θ)]1−ϱ(gψ)(ν)dν=Γ(ϱ)[ψ(θ+ζ2)Iϱ(θ+ζ2)−;pg(θ)−Iϱ(θ+ζ2)−;p(gψ)(θ)]. |
Similarly, integrating I2 by parts, we get
I2=Γ(ϱ)[ψ(θ+ζ2)Iϱ(θ+ζ2)+;pg(ζ)−Iϱ(θ+ζ2)+;p(gψ)(ζ)]. |
Therefore, by substituting the values of I1 and I2 in (2.1), we get the desired result.
Corollary 2.2. If we choose p(ν)=ν in Lemma 2.1, then it reduce to [18,Lemma 4], i.e.,
ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]−[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]=1Γ(ϱ)ζ∫θκ(ν)ψ′(ν)dν, |
where
κ(t)={t∫θ(u−θ)ϱ−1g(u)du,t∈[θ,θ+ζ2],t∫ζ(ζ−u)ϱ−1g(u)du, t∈[θ+ζ2,ζ]. |
Theorem 2.3. Let ϱ>0 and p:[θ,ζ]→R be an increasing and positive function on (θ,ζ], having a continuous derivative p′ on (θ,ζ). Let g:[θ,ζ]→R be nonnegative, integrable and ψ is a convex function on [θ,ζ], then the following Hermite-Hadamard-Fejer type inequality for generalized fractional integrals holds.
Ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]≤[Iϱ(θ+ζ2)+;pgΨ(ζ)+Iϱ(θ+ζ2)−;pgΨ(θ)]≤Ψ(θ)+Ψ(ζ)2[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)], | (2.2) |
where Ψ is defined as Ψ(ν)=ψ(ν)+˜ψ(ν) and ˜ψ(ν)=ψ(θ+ζ−ν).
Proof. Since ψ is convex on [θ,ζ], for all x,y∈[θ,ζ], we have
ψ(x+y2)≤ψ(x)+ψ(y)2. |
Now, for ν∈[0,1], choose x=ν2θ+2−ν2ζ and y=2−ν2θ+ν2ζ, we get
2ψ(θ+ζ2)≤ψ(ν2θ+2−ν2ζ)+ψ(2−ν2θ+ν2ζ). | (2.3) |
Multiplying both sides of (2.3) by
ζ−θ2Γ(ϱ)p′(ν2θ+2−ν2ζ)g(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱ, |
and integrating the resulting inequality with respect to ν over [0,1], we get
ζ−θ2Γ(ϱ)Ψ(θ+ζ2)∫10p′(ν2θ+2−ν2ζ)g(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱdν≤ζ−θ2Γ(ϱ)∫10p′(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱg(ν2θ+2−ν2ζ)ψ(ν2θ+2−ν2ζ)dν+ζ−θ2Γ(ϱ)∫10p′(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱg(ν2θ+2−ν2ζ)ψ(2−ν2θ+ν2ζ)dν. |
Substituting u=ν2θ+2−ν2ζ, we obtain
Ψ(θ+ζ2)Iϱ(θ+ζ2)+;pg(ζ)≤Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)+;pg˜ψ(ζ), |
i.e.
Ψ(θ+ζ2)Iϱ(θ+ζ2)+;pg(ζ)≤Iϱ(θ+ζ2)+;pgΨ(ζ). | (2.4) |
Similarly, multiplying both sides of (2.3) by
ζ−θ2Γ(ϱ)p′(ν2θ+2−ν2ζ)g(ν2θ+2−ν2ζ)(p(ν2θ+2−ν2ζ)−p(θ))1−ϱ, |
and integrating the resulting inequality with respect to ν over [0,1], we obtain
Ψ(θ+ζ2)Iϱ(θ+ζ2)−;pg(θ)≤Iϱ(θ+ζ2)−;pgΨ(θ). | (2.5) |
By adding the inequalities (2.4) and (2.5), we have
Ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]≤[Iϱ(θ+ζ2)+;pgΨ(ζ)+Iϱ(θ+ζ2)−;pgΨ(θ)], | (2.6) |
which prove the left half part of inequality (2.2).
For the proof of second half, since ψ is convex on [θ,ζ], we have
ψ(ν2θ+2−ν2ζ)+ψ(2−ν2θ+ν2ζ)≤ψ(θ)+ψ(ζ). | (2.7) |
Multiplying both sides of (2.7) by
ζ−θ2Γ(ϱ)p′(ν2θ+2−ν2ζ)g(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱ, |
and integrating the resulting inequality with respect to ν over [0,1], we get
ζ−θ2Γ(ϱ)∫10p′(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱg(ν2θ+2−ν2ζ)ψ(ν2θ+2−ν2ζ)dν+ζ−θ2Γ(ϱ)∫10p′(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱg(ν2θ+2−ν2ζ)ψ(2−ν2θ+ν2ζ)dν≤[ψ(θ)+ψ(ζ)]ζ−θ2Γ(ϱ)∫10p′(ν2θ+2−ν2ζ)g(ν2θ+2−ν2ζ)(p(ζ)−p(ν2θ+2−ν2ζ))1−ϱdν. |
Using the change of variable u=ν2θ+2−ν2ζ, we get
Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)+;pg˜ψ(ζ)≤[ψ(θ)+ψ(ζ)]Iϱ(θ+ζ2)+;pg(ζ), |
that is
Iϱ(θ+ζ2)+;pgΨ(ζ)≤Ψ(θ)+Ψ(ζ)2Iϱ(θ+ζ2)+;pg(ζ). | (2.8) |
Similarly, multiplying both sides of (2.7) by
ζ−θ2Γ(ϱ)p′(2−ν2θ+ν2ζ)g(2−ν2θ+ν2ζ)(p(ζ)−p(2−ν2θ+ν2ζ))1−ϱ, |
and integrating the resulting inequality with respect to ν over [0,1], we get
Iϱ(θ+ζ2)−;pgΨ(θ)≤Ψ(θ)+Ψ(ζ)2Iϱ(θ+ζ2)−;pg(θ). | (2.9) |
By adding (2.8) and (2.9), we obtain
[Iϱ(θ+ζ2)+;pgΨ(ζ)+Iϱ(θ+ζ2)−;pgΨ(θ)]≤Ψ(θ)+Ψ(ζ)2[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]. | (2.10) |
From (2.6) and (2.10), we get inequality (2.2).
Corollary 2.4. If we take p(ν)=ν and g is symmetric to ζ+θ2 in Theorem 2.3, then it reduces to the following inequality
ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]≤[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]≤ψ(θ)+ψ(ζ)2[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)], |
for RL fractional integral.
Theorem 2.5. Let p:[θ,ζ]→R be an increasing and positive function on (θ,ζ], having a continuous derivative p′ on (θ,ζ). Let ψ:I→R be a differentiable function on I0 and ψ′∈L[θ,ζ] and g:[θ,ζ]→R is continuous function. If |ψ′|∈K2s on [θ,ζ] for some fixed s∈(0,1], then the following inequality for fractional integrals holds.
|Ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgΨ(ζ)+Iϱ(θ+ζ2)−;pgΨ(θ)]|≤‖g‖∞,[θ,ζ](|ψ′(θ)|+|ψ′(ζ)|)(ζ−θ)sΓ(ϱ+1)[(θ+ζ2∫θ[p(ν)−p(θ)]ϱ+ζ∫θ+ζ2[p(ζ)−p(ν)]ϱ)×((ζ−ν)s+(ν−θ)s)dν], |
where Ψ is defined as Ψ(ν)=ψ(ν)+˜ψ(ν) and ˜ψ(ν)=ψ(θ+ζ−ν).
Proof. By using the pre define properties, we can write
|Ψ′(ν)|=|ψ′(ν)−ψ′(θ+ζ−ν)|≤|ψ′(ν)|+|ψ′(θ+ζ−ν)|=|ψ′(ζ−νζ−θθ+ν−θζ−θζ)|+|ψ′(ν−θζ−θθ+ζ−νζ−θζ)|≤(ζ−νζ−θ)s|ψ′(θ)|+(ν−θζ−θ)s|ψ′(ζ)|+(ν−θζ−θ)s|ψ′(θ)|+(ζ−νζ−θ)s|ψ′(ζ)|=((ζ−νζ−θ)s+(ν−θζ−θ)s)(|ψ′(θ)|+|ψ′(ζ)|). | (2.11) |
By using Lemma 2.1 and inequality (2.11), we get
|Ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgΨ(ζ)+Iϱ(θ+ζ2)−;pgΨ(θ)]|≤1Γ(ϱ)ζ∫θ|κ(ν)||Ψ′(ν)|dν≤1Γ(ϱ)(θ+ζ2∫θ|ν∫θp′(u)[p(u)−p(θ)]1−ϱg(u)du|+ζ∫θ+ζ2|ν∫ζp′(u)[p(ζ)−p(u)]1−ϱg(u)du|)|Ψ′(ν)|dν≤‖g‖∞,[θ,ζ]Γ(ϱ)(θ+ζ2∫θ|ν∫θp′(u)[p(u)−p(θ)]1−ϱdu|+ζ∫θ+ζ2|ν∫ζp′(u)[p(ζ)−p(u)]1−ϱdu|)|Ψ′(ν)|dν≤‖g‖∞,[θ,ζ]Γ(ϱ)[(θ+ζ2∫θ[p(ν)−p(θ)]ϱϱ+ζ∫θ+ζ2[p(ζ)−p(ν)]ϱϱ)×((ζ−νζ−θ)s+(ν−θζ−θ)s)(|ψ′(θ)|+|ψ′(ζ)|)dν]=‖g‖∞,[θ,ζ](|ψ′(θ)|+|ψ′(ζ)|)(ζ−θ)sΓ(ϱ+1)[(θ+ζ2∫θ[p(ν)−p(θ)]ϱ+ζ∫θ+ζ2[p(ζ)−p(ν)]ϱ)×((ζ−ν)s+(ν−θ)s)dν]. |
The proof is done.
Corollary 2.6. If we take s=1 in Theorem 2.5, then we have
|Ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgΨ(ζ)+Iϱ(θ+ζ2)−;pgΨ(θ)]|≤‖g‖∞,[θ,ζ](|ψ′(θ)|+|ψ′(ζ)|)Γ(ϱ+1){θ+ζ2∫θ[p(ν)−p(θ)]ϱdν+ζ∫θ+ζ2[p(ζ)−p(ν)]ϱdν}. |
Corollary 2.7. If we take s=1, p(ν)=ν and g is symmetric to θ+ζ2 in Theorem 2.5, then we get [18,Theorem 6], i.e.,
|ψ(θ+ζ2)[Iϱ(θ+ζ2)−g(θ)+Iϱ(θ+ζ2)+g(ζ)]−[Iϱ(θ+ζ2)−ψg(θ)+Iϱ(θ+ζ2)+ψg(ζ)]|≤‖g‖∞,[θ,ζ](ζ−θ)ϱ+1(ϱ+1)Γ(ϱ+1)2ϱ+1(|ψ′(θ)|+|ψ′(ζ)|). |
Corollary 2.8. If we choose p(ν)=ν and g is symmetric to θ+ζ2 in Theorem 2.5, then we get [19,Theorem 4]
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]−[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)ϱ+1(|ψ′(θ)|+|ψ′(ζ)|)Γ(ϱ+1)[B12(ϱ+1,s+1)+12ϱ+s+1(ϱ+s+1)]. |
Theorem 2.9. Let p:[θ,ζ]→R be an increasing and positive function on (θ,ζ], having a continuous derivative p′ on (θ,ζ). Let ψ:I→R be a differentiable function on I0 and ψ′∈L[θ,ζ] and g:[θ,ζ]→R is continuous function. If |ψ′|q∈K2s on [θ,ζ] for some fixed s∈(0,1] and q≥1, then the following inequality for fractional integrals holds.
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)sqΓ(ϱ+1){(θ+ζ2∫θ[p(ν)−p(θ)]ϱdν)1−1q×(θ+ζ2∫θ[p(ν)−p(θ)]ϱ((ζ−ν)s|ψ′(θ)|q+(ν−θ)s|ψ′(ζ)|q)dν)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱdν)1−1q×(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱ((ζ−ν)s|ψ′(θ)|q+(ν−θ)s|ψ′(ζ)|q)dν)1q}. |
Proof. By Lemma 2.1, Power mean inequality and Definition 1.2, we have
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]|≤1Γ(ϱ)ζ∫θ|κ(ν)||ψ′(ν)|dν≤1Γ(ϱ)(ζ∫θ|κ(ν)|dν)1−1q(ζ∫θ|κ(ν)||ψ′(ν)|qdν)1q≤‖g‖∞,[θ,θ+ζ2]Γ(ϱ){(θ+ζ2∫θ|ν∫θp′(u)[p(u)−p(θ)]ϱ−1du|dν)1−1q×(θ+ζ2∫θ|ν∫θp′(u)[p(u)−p(θ)]ϱ−1du|((ζ−νζ−θ)s|ψ′(θ)|q+(ν−θζ−θ)s|ψ′(ζ)|q)dν)1q}+‖g‖∞,[θ+ζ2,ζ]Γ(ϱ){(ζ∫θ+ζ2|ν∫ζp′(u)[p(ζ)−p(u)]ϱ−1du|dν)1−1q×(ζ∫θ+ζ2|ζ∫νp′(u)[p(ζ)−p(u)]ϱ−1du|((ζ−νζ−θ)s|ψ′(θ)|q+(ν−θζ−θ)s|ψ′(ζ)|q)dν)1q}=‖g‖∞,[θ,ζ](ζ−θ)sqΓ(ϱ+1){(θ+ζ2∫θ[p(ν)−p(θ)]ϱdν)1−1q×(θ+ζ2∫θ[p(ν)−p(θ)]ϱ((ζ−ν)s|ψ′(θ)|q+(ν−θ)s|ψ′(ζ)|q)dν)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱdν)1−1q×(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱ((ζ−ν)s|ψ′(θ)|q+(ν−θ)s|ψ′(ζ)|q)dν)1q}. |
The proof is done.
Corollary 2.10. If we choose s=1 in Theorem 2.9, we get
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)1qΓ(ϱ+1){(θ+ζ2∫θ[p(ν)−p(θ)]ϱdν)1−1q×(θ+ζ2∫θ[p(ν)−p(θ)]ϱ((ζ−ν)|ψ′(θ)|q+(ν−θ)|ψ′(ζ)|q)dν)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱdν)1−1q×(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱ((ζ−ν)|ψ′(θ)|q+(ν−θ)|ψ′(ζ)|q)dν)1q}. |
Corollary 2.11. If we choose s=1, and p(ν)=ν in Theorem 2.9, then it reduces to [18,Theorem 7], i.e.,
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]−[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)ϱ+12ϱ+1+1q(ϱ+1)(ϱ+2)1qΓ(ϱ+1){((ϱ+3)|ψ′(θ)|q+(ϱ+1)|ψ′(ζ)|q)1q+((ϱ+1)|ψ′(θ)|q+(ϱ+3)|ψ′(ζ)|q)1q}. |
Corollary 2.12. If we take p(ν)=ν in Theorem 2.9, then it reduce to [19,Theorem 5], that is
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]−[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)ϱ+12ϱ+1+1q(ϱ+1)(ϱ+s+1)1qΓ(ϱ+1)×[(2ϱ+1(ϱ+1)(ϱ+s+1)B12(ϱ+1,s+1)|ψ′(θ)|q+21−s(ϱ+1)|ψ′(ζ)|q)1q+(21−s(ϱ+1)|ψ′(θ)|q+2ϱ+1(ϱ+1)(ϱ+s+1)B12(ϱ+1,s+1)|ψ′(ζ)|q)1q]. |
Theorem 2.13. Let p:[θ,ζ]→R be an increasing and positive function on (θ,ζ], having a continuous derivative p′ on (θ,ζ). Let ψ:I→R be a differentiable function on I0 and ψ′∈L[θ,ζ] and g:[θ,ζ]→R is continuous function. If |ψ′|q∈K2s on [θ,ζ] for some fixed s∈(0,1] and q>1, then the following inequality for fractional integrals holds.
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)1q2sq+1q(s+1)1qΓ(ϱ+1){(θ+ζ2∫θ[p(ν)−p(θ)]ϱpdν)1p((2s+1−1)|ψ′(θ)|q+|ψ′(ζ)|q)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱpdν)1p(|ψ′(θ)|q+(2s+1−1)|ψ′(ζ)|q)1q}. |
Proof. By Lemma 2.1, Hölder's inequality and Definition 1.2, we obtain
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]|≤1Γ(ϱ)ζ∫θ|κ(ν)||ψ′(ν)|dν≤1Γ(ϱ){(θ+ζ2∫θ|κ(ν)|pdν)1p(θ+ζ2∫θ|ψ′(ν)|qdν)1q+(ζ∫θ+ζ2|κ(ν)|pdν)1p(ζ∫θ+ζ2|ψ′(ν)|qdν)1q}≤‖g‖∞,[θ,ζ]Γ(ϱ){(θ+ζ2∫θ|ν∫θp′(u)[p(u)−p(θ)]1−ϱdu|pdν)1p×(θ+ζ2∫θ((ζ−νζ−θ)s|ψ′(θ)|q+(ν−θζ−θ)s|ψ′(ζ)|q)dν)1q+(ζ∫θ+ζ2|ζ∫νp′(u)[p(ζ)−p(u)]1−ϱdu|pdν)1p×(ζ∫θ+ζ2((ζ−νζ−θ)s|ψ′(θ)|q+(ν−θζ−θ)s|ψ′(ζ)|q)dν)1q}=‖g‖∞,[θ,ζ](ζ−θ)sqϱpΓ(ϱ){(θ+ζ2∫θ[p(ν)−p(θ)]ϱpdν)1p((ζ−θ)s+1(2s+1−1)2s+1(s+1)|ψ′(θ)|q+(ζ−θ)s+12s+1(s+1)|ψ′(ζ)|q)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱpdν)1p×((ζ−θ)s+12s+1(s+1)|ψ′(θ)|q+(ζ−θ)s+1(2s+1−1)2s+1(s+1)|ψ′(ζ)|q)1q}=‖g‖∞,[θ,ζ](ζ−θ)1q2sq+1q(s+1)1qΓ(ϱ+1){(θ+ζ2∫θ[p(ν)−p(θ)]ϱpdν)1p((2s+1−1)|ψ′(θ)|q+|ψ′(ζ)|q)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱpdν)1p(|ψ′(θ)|q+(2s+1−1)|ψ′(ζ)|q)1q}. |
That completes the proof of the result.
Corollary 2.14. If we choose s=1 in Theorem 2.13, we have
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+;pg(ζ)+Iϱ(θ+ζ2)−;pg(θ)]−[Iϱ(θ+ζ2)+;pgψ(ζ)+Iϱ(θ+ζ2)−;pgψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)1q23qΓ(ϱ+1)(θ+ζ2∫θ[p(ν)−p(θ)]ϱpdν)1p(3|ψ′(θ)|q+|ψ′(ζ)|q)1q+(ζ∫θ+ζ2[p(ζ)−p(ν)]ϱpdν)1p(|ψ′(θ)|q+3|ψ′(ζ)|q)1q, |
Corollary 2.15. If we take s=1, and p(ν)=ν in Theorem 2.13, then it reduces to [18,Theorem 8], that is
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]−[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)ϱ+12ϱ+1+2q(ϱp+1)1pΓ(ϱ+1)[(3|ψ′(θ)|q+|ψ′(ζ)|q)1q+(|ψ′(θ)|q+3|ψ′(ζ)|q)1q], |
where 1p+1q=1.
Corollary 2.16. If we take p(ν)=ν in Theorem 2.13, then we get [19,Theorem 6], that is
|ψ(θ+ζ2)[Iϱ(θ+ζ2)+g(ζ)+Iϱ(θ+ζ2)−g(θ)]−[Iϱ(θ+ζ2)+gψ(ζ)+Iϱ(θ+ζ2)−gψ(θ)]|≤‖g‖∞,[θ,ζ](ζ−θ)ϱ+12ϱ+1+sq(s+1)1q(ϱp+1)1pΓ(ϱ+1){((2s+1−1)|ψ′(θ)|q+|ψ′(ζ)|q)1q+(|ψ′(θ)|q+(2s+1−1)|ψ′(ζ)|q)1q}, |
where 1p+1q=1.
This section consists of some particular inequalities which generalizes some classical results such like mid-point inequality.
Proposition 3.1. (Mid-point inequality). By using the assumptions ϱ=1, g(ν)=1 and p(ν)=ν in Corollary 2.6, we get the following inequality
|(ζ−θ)ψ(θ+ζ2)−ζ∫θψ(ν)dν|≤(ζ−θ)28(|ψ′(θ)|+|ψ′(ζ)|). |
Proposition 3.2. (Mid-point inequality). By using the assumptions ϱ=1 and p(ν)=ν in Corollary 2.10, we get the following inequality
|ψ(θ+ζ2)ζ∫θg(ν)dν−ζ∫θ(ψg)(ν)dν|≤‖g‖(ζ−θ)2(3)1q8[(2|ψ′(θ)|q+|ψ′(ζ)|q)1q+(|ψ′(θ)|q+2|ψ′(ζ)|q)1q]. |
Proposition 3.3. By using the assumptions ϱ=1 and p(ν)=ν in Corollary 2.14, we get the following inequality
|ψ(θ+ζ2)ζ∫θg(ν)dν−ζ∫θ(ψg)(ν)dν|≤‖g‖(ζ−θ)2(p+1)1p24p−2p[(3|ψ′(θ)|q+|ψ′(ζ)|q)1q+(|ψ′(θ)|q+3|ψ′(ζ)|q)1q]. |
In the present article, we aim to design the generalized inequalities for RL fractional integral of a function concerning the other function. For this purpose, we use the classical convex and s-convex mappings and develop several inequalities. This work includes equality so that, we can make progress in finding more inequalities by using different types of functions in equality. The findings of these investigations complement those of previous studies. Simply, the recent study confirms the earlier results and play an additional role by making generalizations.
The authors declares that there is no conflict of interests regarding the publication of this paper.
[1] |
Y. Cho, I. Kim, D. Sheen, A fractional-order model for MINMOD Millennium, Math. Biosci., 262 (2015), 36–45. doi: 10.1016/j.mbs.2014.11.008
![]() |
[2] |
R. L. Magin, M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, J. Vib. Control, 14 (2008), 1431–1442. doi: 10.1177/1077546307087439
![]() |
[3] |
W. Chen, G. Pang, A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction, J. Comput. Phys., 309 (2016), 350–367. doi: 10.1016/j.jcp.2016.01.003
![]() |
[4] |
N. M. Grahovac, M. M. Zigic, Modelling of the hamstring muscle group by use of fractional derivatives, Comput. Math. Appl., 59 (2010), 1695–1700. doi: 10.1016/j.camwa.2009.08.011
![]() |
[5] | Y. A. Rossikhin, M. V. Shitikova, Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts, Cent. Eur. J. Phys., 11 (2013), 760–778. |
[6] |
M. A. Balci, Fractional virus epidemic model on financial networks, Open Math., 14 (2016), 1074–1086. doi: 10.1515/math-2016-0098
![]() |
[7] |
H. E. Roman, M. Porto, Fractional derivatives of random walks: Time series with long-time memory, Phys. Rev. E, 78 (2008), 031127. doi: 10.1103/PhysRevE.78.031127
![]() |
[8] |
M. A. Moreles, R. Lainez, Mathematical modelling of fractional order circuit elements and bioimpedance applications, Commun. Nonlinear Sci. Numer. Simul., 46 (2017), 81–88. doi: 10.1016/j.cnsns.2016.10.020
![]() |
[9] |
E. Reyes-Melo, J. Martinez-Vega, C. Guerrero-Salazar, U. Ortiz-Mendez, Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials, J. Appl. Polymer Sci., 98 (2005), 923–935. doi: 10.1002/app.22057
![]() |
[10] |
J. M. Carcione, Theory and modeling of constant-QP- and S-waves using fractional time derivatives, Geophysics, 74 (2009), 1–11. doi: 10.1190/1.3273876
![]() |
[11] | J. L. W. V. Jensen, Sur les fonctions convexes et les inegalites entre les valeurs moyennes, Acta Math., 30 (1905), 175–193. |
[12] | W. W. Breckner, Stetigkeitsaussagen für eine Klasse verallgemein konvexer Funktionen in topologischen linearen Räumen, Publ. Inst. Math., 23 (1978), 13–20. |
[13] | I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals Series and Products, 7 Eds., San Diego: Academic Press, 1980. |
[14] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Switzerland: Gordon and Breach, 1993. |
[15] |
H. Chen, U. N. Katugampola, Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integrals, J. Math. Anal. Appl., 446 (2017), 1274–1291. doi: 10.1016/j.jmaa.2016.09.018
![]() |
[16] |
T. U. Khan, M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. doi: 10.1016/j.cam.2018.07.018
![]() |
[17] |
F. Jarad, E. Ugurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Differ. Equ., 2017 (2017), 1–16. doi: 10.1186/s13662-016-1057-2
![]() |
[18] | E. Set, I. Iscan, M. Z. Sarikaya, M. E. Ozdemir, On new inequalities of Hermite-Hadamard-Fejèr type for convex functions via fractional integrals, Appl. Math. Comput., 259 (2015), 875–881. |
[19] |
M. T. Hakiki, A. Wibowo, Hermite-Hadamard-Fejèr type inequalities for s-convex functions in the second sense via Riemann-Liouville fractional integral, J. Phys. Conf. Ser., 1442 (2020), 012039. doi: 10.1088/1742-6596/1442/1/012039
![]() |
1. | Tao Yan, Ghulam Farid, Ayşe Kübra Demirel, Kamsing Nonlaopon, Sun Young Cho, Further on Inequalities for α , h − m -Convex Functions via k -Fractional Integral Operators, 2022, 2022, 2314-4785, 1, 10.1155/2022/9135608 | |
2. | Muhammad Samraiz, Kanwal Saeed, Saima Naheed, Gauhar Rahman, Kamsing Nonlaopon, On inequalities of Hermite-Hadamard type via $ n $-polynomial exponential type $ s $-convex functions, 2022, 7, 2473-6988, 14282, 10.3934/math.2022787 | |
3. | Muhammad Samraiz, Muhammad Umer, Artion Kashuri, Thabet Abdeljawad, Sajid Iqbal, Nabil Mlaiki, On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation, 2021, 5, 2504-3110, 118, 10.3390/fractalfract5030118 | |
4. | Peng Xu, Saad Ihsan Butt, Saba Yousaf, Adnan Aslam, Tariq Javed Zia, Generalized Fractal Jensen–Mercer and Hermite–Mercer type inequalities via h-convex functions involving Mittag–Leffler kernel, 2022, 61, 11100168, 4837, 10.1016/j.aej.2021.10.033 | |
5. | YONGFANG QI, QINGZHI WEN, GUOPING LI, KECHENG XIAO, SHAN WANG, DISCRETE HERMITE–HADAMARD-TYPE INEQUALITIES FOR (s,m)-CONVEX FUNCTION, 2022, 30, 0218-348X, 10.1142/S0218348X22501602 | |
6. | Muhammad Tariq, Sotiris K. Ntouyas, Asif Ali Shaikh, A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators, 2023, 12, 2075-1680, 719, 10.3390/axioms12070719 | |
7. | Miguel Vivas-Cortez, Muzammil Mukhtar, Iram Shabbir, Muhammad Samraiz, Muhammad Yaqoob, On Fractional Integral Inequalities of Riemann Type for Composite Convex Functions and Applications, 2023, 7, 2504-3110, 345, 10.3390/fractalfract7050345 | |
8. | Miguel Vivas-Cortez, Muhammad Samraiz, Muhammad Tanveer Ghaffar, Saima Naheed, Gauhar Rahman, Yasser Elmasry, Exploration of Hermite–Hadamard-Type Integral Inequalities for Twice Differentiable h-Convex Functions, 2023, 7, 2504-3110, 532, 10.3390/fractalfract7070532 | |
9. | Muhammad Samraiz, Maria Malik, Saima Naheed, Gauhar Rahman, Kamsing Nonlaopon, Hermite–Hadamard-type inequalities via different convexities with applications, 2023, 2023, 1029-242X, 10.1186/s13660-023-02957-7 | |
10. | Muhammad Samraiz, Saima Naheed, Ayesha Gul, Gauhar Rahman, Miguel Vivas-Cortez, Innovative Interpolating Polynomial Approach to Fractional Integral Inequalities and Real-World Implementations, 2023, 12, 2075-1680, 914, 10.3390/axioms12100914 | |
11. | Miguel Vivas-Cortez, Muhammad Samraiz, Aman Ullah, Sajid Iqbal, Muzammil Mukhtar, A modified class of Ostrowski-type inequalities and error bounds of Hermite–Hadamard inequalities, 2023, 2023, 1029-242X, 10.1186/s13660-023-03035-8 | |
12. | XIAOHUA ZHANG, YUNXIU ZHOU, TINGSONG DU, PROPERTIES AND 2α̃-FRACTAL WEIGHTED PARAMETRIC INEQUALITIES FOR THE FRACTAL (m,h)-PREINVEX MAPPINGS, 2023, 31, 0218-348X, 10.1142/S0218348X23501347 |