As we know, Atangana and Baleanu developed great fractional integral operators which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Inspired by these integral operators, we derive in this paper two new fractional integral identities involving Atangana-Baleanu fractional integrals. Using these identities as auxiliary results, we establish new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly n-polynomial convex functions. We also discuss several important special cases of the main results. In order to show the efficiency of our main results, we offer applications for special means and for differentiable functions of first and second order that are in absolute value bounded.
Citation: Miguel Vivas-Cortez, Muhammad Uzair Awan, Sehrish Rafique, Muhammad Zakria Javed, Artion Kashuri. Some novel inequalities involving Atangana-Baleanu fractional integral operators and applications[J]. AIMS Mathematics, 2022, 7(7): 12203-12226. doi: 10.3934/math.2022678
[1] | Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo, Artion Kashuri, Taher A. Nofal, Ching-Hsien Hsu . Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications. AIMS Mathematics, 2022, 7(8): 15159-15181. doi: 10.3934/math.2022831 |
[2] | Soubhagya Kumar Sahoo, Fahd Jarad, Bibhakar Kodamasingh, Artion Kashuri . Hermite-Hadamard type inclusions via generalized Atangana-Baleanu fractional operator with application. AIMS Mathematics, 2022, 7(7): 12303-12321. doi: 10.3934/math.2022683 |
[3] | Saad Ihsan Butt, Erhan Set, Saba Yousaf, Thabet Abdeljawad, Wasfi Shatanawi . Generalized integral inequalities for ABK-fractional integral operators. AIMS Mathematics, 2021, 6(9): 10164-10191. doi: 10.3934/math.2021589 |
[4] | Tekin Toplu, Mahir Kadakal, İmdat İşcan . On n-Polynomial convexity and some related inequalities. AIMS Mathematics, 2020, 5(2): 1304-1318. doi: 10.3934/math.2020089 |
[5] | M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253 |
[6] | Hong Yang, Shahid Qaisar, Arslan Munir, Muhammad Naeem . New inequalities via Caputo-Fabrizio integral operator with applications. AIMS Mathematics, 2023, 8(8): 19391-19412. doi: 10.3934/math.2023989 |
[7] | Haoliang Fu, Muhammad Shoaib Saleem, Waqas Nazeer, Mamoona Ghafoor, Peigen Li . On Hermite-Hadamard type inequalities for $ n $-polynomial convex stochastic processes. AIMS Mathematics, 2021, 6(6): 6322-6339. doi: 10.3934/math.2021371 |
[8] | Jia-Bao Liu, Saad Ihsan Butt, Jamshed Nasir, Adnan Aslam, Asfand Fahad, Jarunee Soontharanon . Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator. AIMS Mathematics, 2022, 7(2): 2123-2141. doi: 10.3934/math.2022121 |
[9] | 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 |
[10] | Jamshed Nasir, Saber Mansour, Shahid Qaisar, Hassen Aydi . Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel. AIMS Mathematics, 2023, 8(5): 10001-10020. doi: 10.3934/math.2023506 |
As we know, Atangana and Baleanu developed great fractional integral operators which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Inspired by these integral operators, we derive in this paper two new fractional integral identities involving Atangana-Baleanu fractional integrals. Using these identities as auxiliary results, we establish new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly n-polynomial convex functions. We also discuss several important special cases of the main results. In order to show the efficiency of our main results, we offer applications for special means and for differentiable functions of first and second order that are in absolute value bounded.
A set K⊂R is said to be convex, if ∀Θ,Ω∈K and τ∈[0,1], we have
(1−τ)Θ+τΩ∈K. |
A function Φ:K→R is said to be convex, if ∀Θ,Ω∈K and τ∈[0,1], we get
Φ((1−τ)Θ+τΩ)≤(1−τ)Φ(Θ)+τΦ(Ω). |
Theory of convexity has also played an important part in the development of theory of inequalities. Several results in theory of inequalities are direct consequences of the applications of convexity. Among these results one of the most extensively as well as intensively studied result is the Hermite-Hadamard inequality. This result reads as:
Let Φ:I=[a1,a2]⊆R→R be a convex function, then
Φ(a1+a22)≤1a2−a1a2∫a1Φ(Θ)dΘ≤Φ(a1)+Φ(a2)2. |
In recent years, different approaches have been used in obtaining new versions of Hermite-Hadamard inequality. Mohammed et al. [1] found new discrete inequalities of Hermite-Hadamard type for convex functions. Mohammed et al. [2] established generalized Hermite-Hadamard inequalities via the tempered fractional integrals. Mohammed et al. [3] obtained new fractional inequalities of Hermite-Hadamard type involving the incomplete gamma functions. Rahman et al. [4] derived certain fractional proportional integral inequalities via convex functions. Rahman et al. [5] give new bounds of generalized proportional fractional integrals in general form via convex functions and presented their applications. Sarikaya et al. [6] have used the approach of fractional calculus and obtained fractional analogues of Hermite-Hadamard inequality. Since then a variety of different approaches from fractional calculus have been used in obtaining fractional analogues of classical inequalities. For instance, Budak and Agarwal [7] obtained new generalized fractional midpoint type inequalities. Chu et al. [8] obtained new Simpson type of inequalities using Katugampola fractional integrals. Chu et al. [9] new generalized fractional Hermite-Hadamard inequality using χk-Hilfer fractional integrals. Kashuri et al. [10] obtained new generalized fractional integral identities and obtained new inequalities. Liu et al. [11] obtained new Hermite-Hadamard type of inequalities via ψ-fractional integrals. Onalan et al. [12] obtained fractional analogues of Hermite-Hadamard type integral inequalities via fractional integral operators with Mittag-Leffler kernel. Talib and Awan [13] obtained some new estimates of upper bounds for n-th order differentiable functions involving χ-Riemann-Liouville integrals via γ-preinvex functions. Wu et al. [14] obtained estimates of upper bound for a k-th order differentiable functions involving Riemann-Liouville integrals via higher order strongly h-preinvex functions. Wu et al. [15] established some integral inequalities for n-polynomial ζ-preinvex functions. Zhang et al. [16] obtained new k-fractional integral inequalities containing multi parameters via generalized (s,m)-preinvexity property of the functions. Huang et al. [17] derived some inequalities of the Hermite-Hadamard type via k-fractional conformable integrals. Rahman et al. [18] established certain inequalities via generalized proportional Hadamard fractional integral operators. Rahman et al. [19] obtained the Minkowski inequalities via generalized proportional fractional integral operators. For some more details, see [20,21,22,23,24,25].
In recent years, several new extensions and generalizations of classical convex functions have been defined in the literature. Chu et al. [8] introduced the notion of higher order strongly n-polynomial convex function as follows:
Definition 1.1. [[8]] A function Φ:K→R is said to be higher order strongly n-polynomial convex, if ∀Θ,Ω∈K,τ∈[0,1], and u,σ>0, we have
Φ(τΘ+(1−τ)Ω)≤1nn∑s=1[1−(1−τ)s]Φ(Θ)+1nn∑s=1[1−τs]Φ(Ω)−u(τσ(1−τ)+τ(1−τ)σ)‖Ω−Θ‖σ. |
Remark 1.1. Note that, if we take u=0 in Definition 1, then we have the class of n-polynomial convex functions introduced and studied by Toplu et al. [26]. If we take σ=2 in Definition 1, then we get the class of strongly n-polynomial convex functions. If we take n=1 and σ=2, then we obtain the class of strongly convex functions [27].
Fractional calculus is an effective tool to explain physical phenomenas and also real world problems. The concept of fractional order derivative and integrals that will shed light on some unknown points about differential equations and solutions of some fractional order differential equations, which proved to be useless for their solution, is a novelty in applied sciences as well as in mathematics. New derivatives and integrals contribute to the solution of differential equations that are expressed and solved in classical analysis, as well as fractional order derivatives and integrals. In addition, it has increased its contribution to the literature with its applications in areas such as engineering, biostatistics and mathematical biology. Fractional derivative and integral operators not only differed from each other in terms of singularity, locality and kernels, but also brought innovations to fractional analysis in terms of their usage areas and spaces. Baleanu et al. [28], investigated existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions. Khan et al. [29], analyzed positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space. Khan et al. [30], give the existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. For more details regarding fractional calculus and their applications, see [31,32,33,34,35].
Let us recall some basic concepts from fractional calculus which will be helpful in obtaining our main results.
Definition 1.2. [36] Let Φ∈L1[a1,a2]. The Riemann-Liouville integrals Jνa1+Φ and Jνa2−Φ of order ν>0 are defined by
Jνa1+Φ(Θ)=1Γ(ν)Θ∫a1(Θ−τ)ν−1Φ(τ)dτ,Θ>a1, |
and
Jνa2−Φ(Θ)=1Γ(ν)a2∫Θ(τ−Θ)ν−1Φ(τ)dτ,Θ<a2. |
In 2015 Caputo and Fabrizio suggested a new operator with fractional order, this derivative is based on the exponential kernel. Earlier this year 2016 Atangana and Baleanu developed another version which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Both operators show some properties of filter. However, the Atangana and Baleanu version has in addition to this, all properties of fractional derivative. This shown effectiveness and advantages of the Atangana-Baleanu integral operators.
So, let see the following definitions about Atangana-Baleanu fractional derivatives and fractional integrals that are given in [37,38,39], respectively.
Definition 1.3. Let Φ∈H1(a1,a2) and not necessarily differentiable then, the definition of the new fractional derivative (Atangana-Baleanu fractional derivative in Riemann-Liouville sense) is given as follows:
ABRa1DατΦ(τ)=B(α)1−αddτ∫τa1Φ(Θ)Eα[−α(τ−Θ)α1−α]dΘ,a2>a1,α∈[0,1], |
where Eα(Θ) is the well-known Mittag-Leffler function.
Definition 1.4. Let Φ∈H1(a1,a2) then, the definition of the new fractional derivative (Atangana-Baleanu derivative in Caputo sense) is given as:
ABCa1DατΦ(τ)=B(α)1−α∫τa1Φ′(Θ)Eα[−α(τ−Θ)α1−α]dΘ,a2>a1,α∈[0,1]. |
Here, B(α)>0 is the normalization function which satisfies the condition B(0)=B(1)=1. They suggested that B(α) has the same properties as in Caputo and Fabrizio case. The above definitions are very helpful to real world problem and also they have great advantages when Laplace transform is apply to solve some physical problems. Since the normalization function B(α) is positive, it immediately follows that the fractional Atangana-Baleanu integral of a positive function is positive. It should be noted that, when the order α→1, we recover the classical integral. Also, the initial function is recovered whenever the fractional order α→0.
Definition 1.5. The left hand side fractional integral related to the new fractional derivative with nonlocal kernel of a function Φ∈L1[a1,a2] is defined as follows:
ABa1IατΦ(τ)=1−αB(α)Φ(τ)+αB(α)Γ(α)∫τa1Φ(Θ)(τ−Θ)α−1dΘ,a2>a1,α∈[0,1]. |
The right hand side of Atangana-Baleanu fractional integral is given as:
ABIαa2Φ(τ)=1−αB(α)Φ(τ)+αB(α)Γ(α)∫a2τΦ(Θ)(Θ−τ)α−1dΘ. |
Here, Γ(α) is the gamma function.
For more details about Atangana-Baleanu fractional integral operators and their applications, see [40,41,42,43,44,45,46,47,48].
Inspired by above results, the main motivation of this paper is to establish two new fractional integral identities involving Atangana-Baleanu fractional integrals. Using these identities as auxiliary results, we will derive new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly n-polynomial convex functions. Furthermore, in order to show the efficiency of our main results, we will offer applications for special means and for differentiable functions of first and second order that are in absolute value bounded. Finally, some conclusions and future research will be given.
In this section, we derive two new fractional integral identities using Atangana-Baleanu fractional integrals.
Lemma 2.1. Let Φ:[a1,a2]→R be a differentiable function on (a1,a2) with a1<a2. If Φ′∈L1[a1,a2], then for all Θ∈[a1,a2], we have
ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]=1B(α)Γ(α)[(Θ−a1)α+1∫10(1−τα)Φ′(τa2+(1−τ)(a1+a2−Θ))dτ+(a2−Θ)α+1∫10(τα−1)Φ′(τa1+(1−τ)(a1+a2−Θ))dτ]. |
Proof. Integrating by parts, we have
∫10(1−τα)Φ′(τa2+(1−τ)(a1+a2−Θ))dτ=−Φ(a1+a2−Θ)Θ−a1+αΘ−a1∫10τα−1Φ(τa2+(1−τ)(a1+a2−Θ))dτ=−Φ(a1+a2−Θ)Θ−a1+α(Θ−a1)α+1∫a2a1+a2−Θ(u−(a1+a2−Θ))α−1Φ(u)du. |
Multiplying both sides of the last inequality by 1B(α)Γ(α), and then adding the term 1−αB(α)Φ(a1+a2−Θ), we get
(Θ−a1)α+1B(α)Γ(α)∫10(1−τα)Φ′(τa2+(1−τ)(a1+a2−Θ))dτ+1−αB(α)Φ(a1+a2−Θ)=−(Θ−a1)αB(α)Γ(α)Φ(a1+a2−Θ)+αB(α)Γ(α)∫a2a1+a2−Θ(u−(a1+a2−Θ))α−1Φ(u)du+1−αB(α)Φ(a1+a2−Θ)=−(Θ−a1)αB(α)Γ(α)Φ(a1+a2−Θ)+ABIαa2{Φ(a1+a2−Θ)}. | (2.1) |
Similarly, we have
(a2−Θ)α+1B(α)Γ(α)∫10(τα−1)Φ′(τa1+(1−τ)(a1+a2−Θ))dτ+1−αB(α)Φ(a1+a2−Θ)=−(a2−Θ)αB(α)Γ(α)Φ(a1+a2−Θ)+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}. | (2.2) |
By the identities (2.1) and (2.2), we obtain the required result.
Remark 2.1. Taking α=1 in Lemma 2.1, we get the following identity:
1a2−a1∫a2a1Φ(Θ)dΘ−Φ(a1+a2−Θ)=1a2−a1[(Θ−a1)2∫10(1−τ)Φ′(τa2+(1−τ)(a1+a2−Θ))dτ+(a2−Θ)2∫10(τ−1)Φ′(τa1+(1−τ)(a1+a2−Θ))dτ]. |
Lemma 2.2. Let Φ:[a1,a2]→R be a twice differentiable function on (a1,a2) with a1<a2. If Φ″∈L1[a1,a2], then for all Θ∈[a1,a2], we have
Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]=1(α+1)B(α)Γ(α)[(Θ−a1)α+2∫10(1−τα+1)Φ″(τa2+(1−τ)(a1+a2−Θ))dτ−(a2−Θ)α+2∫10(τα+1−1)Φ″(τa1+(1−τ)(a1+a2−Θ))dτ]. | (2.3) |
Proof. Consider the right hand side of 2.3, we have
I:=1(α+1)B(α)Γ(α)[(Θ−a1)α+2∫10(1−τα+1)Φ″(τa2+(1−τ)(a1+a2−Θ))dτ−(a2−Θ)α+2∫10(τα+1−1)Φ″(τa1+(1−τ)(a1+a2−Θ))dτ]=1(α+1)B(α)Γ(α)[(Θ−a1)α+2I1−(a2−Θ)α+2I2], | (2.4) |
where
I1:=∫10(1−τα+1)Φ″(τa2+(1−τ)(a1+a2−Θ))dτ=−Φ′(a1+a2−Θ)Θ−a1+α+1Θ−a1∫10ταΦ′(τa2+(1−τ)(a1+a2−Θ))dτ=−Φ′(a1+a2−Θ)Θ−a1+α+1(Θ−a1)[Φ(a2)Θ−a1−α(Θ−a1)∫10τα−1Φ(τa2+(1−τ)(a1+a2−Θ))dτ]=−Φ′(a1+a2−Θ)Θ−a1+(α+1)Φ(a2)(Θ−a1)2−α(α+1)(Θ−a1)2∫10τα−1Φ(τa2+(1−τ)(a1+a2−Θ))dτ=−Φ′(a1+a2−Θ)Θ−a1+(α+1)Φ(a2)(Θ−a1)2+(1+α)Γ(α)Φ(a1+a2−Θ)(Θ−a1)α+2−(1+α)B(α)Γ(α)(Θ−a1)α+2ABIαa2{Φ(a1+a2−Θ)}. |
Similarly, we get
I2:=∫10(τα+1−1)Φ″(τa1+(1−τ)(a1+a2−Θ))dτ=−Φ′(a1+a2−Θ)a2−Θ−(α+1)Φ(a1)(a2−Θ)2−(1+α)Γ(α)Φ(a1+a2−Θ)(a2−Θ)α+2+(1+α)B(α)Γ(α)(a2−Θ)α+2ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}. |
By substituting the values of I1 and I2 in 2.4, we obtain our required result.
Remark 3.2. Taking α=1 in Lemma 2.2, we get the following identity:
Φ′(a1+a2−Θ)[(a2−Θ)2−(Θ−a1)2]+2[(Θ−a1)Φ(a2)+(a2−Θ)Φ(a1)]−2∫a2a1Φ(Θ)dΘ=[(Θ−a1)3∫10(1−τ2)Φ″(τa2+(1−τ)(a1+a2−Θ))dτ−(a2−Θ)3∫10(τ2−1)Φ″(τa1+(1−τ)(a1+a2−Θ))dτ]. |
In this section, we discuss our main results.
Theorem 3.1. Let Φ:[a1,a2]→R be a differentiable function on (a1,a2) with 0≤a1<a2. If |Φ′|q is higher order strongly n-polynomial convex on [a1,a2] for q>1 and 1p+1q=1, then for all Θ∈[a1,a2] the following inequality for fractional integrals holds:
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)(αpαp+1)1p[(Θ−a1)α+1(1nn∑s=1ss+1(|Φ′(a2)|q+|Φ′(a1+a2−Θ)|q)−2u‖Θ−a1‖σ(σ+1)(σ+2))1q+(a2−Θ)α+1(1nn∑s=1ss+1(|Φ′(a1)|q+|Φ′(a1+a2−Θ)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2))1q]. |
Proof. It is evident that
∫10[1−(1−τ)s]dτ=∫10[1−τs]dτ=ss+1. |
Applying Lemma 2.1 and properties of modulus, we have
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)[(Θ−a1)α+1∫10|1−τα||Φ′(τa2+(1−τ)(a1+a2−Θ))|dτ+(a2−Θ)α+1∫10|τα−1||Φ′(τa1+(1−τ)(a1+a2−Θ))|dτ]. |
Using Hölder's inequality and higher order strongly n-polynomial convexity of |Φ′|q, we get
∫10|1−τα||Φ′(τa2+(1−τ)(a1+a2−Θ))|dτ≤(∫10|1−τα|pdτ)1p(∫10|Φ′(τa2+(1−τ)(a1+a2−Θ))|qdτ)1q≤(∫10(1−ταp)dτ)1p(∫10[1nn∑s=1[1−(1−τ)s]|Φ′(a2)|q+1nn∑s=1[1−τs]|Φ′(a1+a2−Θ)|q−u(τσ(1−τ)+τ(1−τ)σ)‖Θ−a1‖σ]dτ)1q=(αpαp+1)1p[1nn∑s=1ss+1(|Φ′(a2)|q+|Φ′(a1+a2−Θ)|q)−2u‖Θ−a1‖σ(σ+1)(σ+2)]1q. | (3.1) |
Here, we use
(A−B)q≤Aq−Bq |
for any A>B≥0 and q≥1.
Similarly, we have
∫10|τα−1||Φ′(τa1+(1−τ)(a1+a2−Θ))|dτ≤(αpαp+1)1p[1nn∑s=1ss+1(|Φ′(a1)|q+|Φ′(a1+a2−Θ)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2)]1q. | (3.2) |
By the inequalities (3.1) and (3.2), we obtain required result.
Corollary 3.1. Taking α=1 in Theorem 3.1 and using Remark 2.1, we have
|Φ(a1+a2−Θ)−1a2−a1∫a2a1Φ(Θ)dΘ|≤1a2−a1(pp+1)1p[(Θ−a1)2(1nn∑s=1ss+1(|Φ′(a2)|q+|Φ′(a1+a2−Θ)|q)−2u‖Θ−a1‖σ(σ+1)(σ+2))1q+(a2−Θ)2(1nn∑s=1ss+1(|Φ′(a1)|q+|Φ′(a1+a2−Θ)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2))1q]. |
Corollary 3.2. Taking u→0+ in Theorem 3.1, we have
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)(αpαp+1)1p[(Θ−a1)α+1(1nn∑s=1ss+1(|Φ′(a2)|q+|Φ′(a1+a2−Θ)|q))1q+(a2−Θ)α+1(1nn∑s=1ss+1(|Φ′(a1)|q+|Φ′(a1+a2−Θ)|q)1q]. |
Corollary 3.3. Under assumptions of Theorem 3.1 with Θ=a1+a22, we have the following inequality
|ABIαa2Φ(a1+a22)+ABa1Iα(a1+a22)Φ(a1+a22)−2Φ(a1+a22)B(α)Γ(α)[(a2−a12)α+(1−α)Γ(α)]|≤1B(α)Γ(α)(αpαp+1)1p(a2−a12)α+1[(1nn∑s=1ss+1(|Φ′(a2)|q+|Φ′(a1+a22)|q)−2u‖(a2−a12)‖σ(σ+1)(σ+2))1q+(1nn∑s=1ss+1(|Φ′(a1)|q+|Φ′(a1+a22)|q)−2u‖(a2−a12)‖σ(σ+1)(σ+2))1q]. |
Theorem 3.2. Let Φ:[a1,a2]→R be a differentiable function on (a1,a2) with 0≤a1<a2. If |Φ′|q is higher order strongly n-polynomial convex on [a1,a2] for q≥1, then for all Θ∈[a1,a2] the following inequality for fractional integrals holds:
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)(αα+1)1−1q[(Θ−a1)α+1[|Φ′(a2)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−1α+1+1α+s+1)−u‖Θ−a1‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ))]1q+(a2−Θ)α+1[|Φ′(a1)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−1α+1+1α+s+1)−u‖a2−Θ‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ))]1q]. |
Proof. From Lemma 2.1, properties of modulus and power mean inequality, we have
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)[(Θ−a1)α+1∫10|1−τα||Φ′(τa2+(1−τ)(a1+a2−Θ))|dτ+(a2−Θ)α+1∫10|τα−1||Φ′(τa1+(1−τ)(a1+a2−Θ))|dτ]≤1B(α)Γ(α)[(Θ−a1)α+1(∫10|1−τα|dτ)1−1q(∫10|1−τα||Φ′(τa2+(1−τ)(a1+a2−Θ))|qdτ)1q+(a2−Θ)α+1(∫10|τα−1|dτ)1−1q(∫10|τα−1||Φ′(τa1+(1−τ)(a1+a2−Θ))|qdτ)1q]. |
Using the higher order strongly n-polynomial convexity of |Φ′|q, we get
∫10|1−τα||Φ′(τa2+(1−τ)(a1+a2−Θ))|dτ≤∫10(1−τα)[1nn∑s=1[1−(1−τ)s]|Φ′(a2)|q+1nn∑s=1[1−τs]|Φ′(a1+a2−Θ)|q−u(τσ(1−τ)+τ(1−τ)σ)‖Θ−a1‖σ]dτ=|Φ′(a2)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−1α+1+1α+s+1)−u‖Θ−a1‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ)). | (3.3) |
Similarly, we have
∫10|τα−1||Φ′(τa1+(1−τ)(a1+a2−Θ))|dτ≤∫10(τα−1)[1nn∑s=1[1−(1−τ)s]|Φ′(a1)|q+1nn∑s=1[1−τs]|Φ′(a1+a2−Θ)|q−u(τσ(1−τ)+τ(1−τ)σ)‖a2−Θ‖σ]dτ=|Φ′(a1)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−1α+1+1α+s+1)−u‖a2−Θ‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ)). | (3.4) |
By the inequalities (3.3) and (3.4), we obtain required result.
Corollary 3.4. Taking α=1 in Theorem 3.2 and using Remark 2.1, we have
|Φ(a1+a2−Θ)−1a2−a1∫a2a1Φ(Θ)dΘ|≤1a2−a1(12)1−1q[(Θ−a1)2[|Φ′(a2)|qnn∑s=1(ss+1−12+Γ(s+1)Γ(s+3))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−12+1s+2)−u‖Θ−a1‖σ(1(σ+1)(σ+2)−1(2+σ)(3+σ)−2Γ(1+σ)Γ(4+σ))]1q+(a2−Θ)2[|Φ′(a1)|qnn∑s=1(ss+1−12+Γ(s+1)Γ(s+3))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−12+1s+2)−u‖a2−Θ‖σ(1(σ+1)(σ+2)−1(2+σ)(3+σ)−2Γ(1+σ)Γ(4+σ))]1q]. |
Corollary 3.5. Taking u→0+ in Theorem 3.2, we have
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)(αα+1)1−1q[(Θ−a1)α+1[|Φ′(a2)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−1α+1+1α+s+1)]1q+(a2−Θ)α+1[|Φ′(a1)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a2−Θ)|qnn∑s=1(ss+1−1α+1+1α+s+1)]1q]. |
Corollary 3.6. Under assumptions of Theorem 3.2 with Θ=a1+a22, we have the following inequality
|ABIαa2Φ(a1+a22)+ABa1Iα(a1+a22)Φ(a1+a22)−2Φ(a1+a22)B(α)Γ(α)[(a2−a12)α+(1−α)Γ(α)]|≤1B(α)Γ(α)(αα+1)1−1q(a2−a12)α+1[[|Φ′(a2)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a22)|qnn∑s=1(ss+1−1α+1+1α+s+1)−u‖(a2−a12)‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ))]1q+[|Φ′(a1)|qnn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+|Φ′(a1+a22)|qnn∑s=1(ss+1−1α+1+1α+s+1)−u‖(a2−a12)‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ))]1q]. |
Theorem 3.3. Let Φ:[a1,a2]→R be a twice differentiable function on (a1,a2) with 0≤a1<a2. If |Φ″|q is higher order strongly n-polynomial convex on [a1,a2] for q>1 and 1p+1q=1, then for all Θ∈[a1,a2] the following inequality for fractional integrals holds:
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)(p(α+1)p(α+1)+1)1p[(Θ−a1)α+2(1nn∑s=1ss+1(|Φ″(a2)|q+|Φ″(a1+a2−Θ)|q)−2u‖Θ−a1‖σ(σ+1)(σ+2))1q+(a2−Θ)α+2(1nn∑s=1ss+1(|Φ″(a1)|q+|Φ″(a1+a2−Θ)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2))1q], |
where 1p+1q=1.
Proof. Applying the Lemma 2.2 and properties of modulus, we have
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)[(Θ−a1)α+2∫10|1−τα+1||Φ″(τa2+(1−τ)(a1+a2−Θ))|dτ+(a2−Θ)α+2∫10|τα+1−1||Φ″(τa1+(1−τ)(a1+a2−Θ))|dτ] |
Using Hölder's inequality and higher order strongly n-polynomial convexity of |Φ″|q, we get
∫10|1−τα+1||Φ″(τa2+(1−τ)(a1+a2−Θ))|dτ≤(∫10|1−τα+1|pdτ)1p(∫10|Φ″(τa2+(1−τ)(a1+a2−Θ))|qdτ)1q≤(∫10(1−τ(α+1)p)dτ)1p(∫10[1nn∑s=1[1−(1−τ)s]|Φ″(a2)|q+1nn∑s=1[1−τs]|Φ″(a1+a2−Θ)|q−u(τσ(1−τ)+τ(1−τ)σ)‖Θ−a1‖σ]dτ)1q=((α+1)p(α+1)p+1)1p[1nn∑s=1ss+1(|Φ″(a2)|q+|Φ″(a1+a2−Θ)|q)−2u‖Θ−a1‖σ(σ+1)(σ+2)]1q. | (3.5) |
Here, we use
(A−B)q≤Aq−Bq |
for any A>B≥0 and q≥1.
Similarly, we have
∫10|τα+1−1||Φ″(τa1+(1−τ)(a1+a2−Θ))|dτ≤((α+1)p(α+1)p+1)1p[1nn∑s=1ss+1(|Φ″(a1)|q+|Φ″(a1+a2−Θ)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2)]1q. | (3.6) |
By the inequalities (3.5) and (3.6), we obtain required result.
Corollary 3.7. Taking α=1 in Theorem 3.3 and using Remark 2.2, we have
|Φ′(a1+a2−Θ)[(a2−Θ)2−(Θ−a1)2]+2[(Θ−a1)Φ(a2)+(a2−Θ)Φ(a1)]−2∫a2a1Φ(Θ)dΘ|≤(2p2p+1)1p[(Θ−a1)3(1nn∑s=1ss+1(|Φ″(a2)|q+|Φ″(a1+a2−Θ)|q)−2u‖Θ−a1‖σ(σ+1)(σ+2))1q+(a2−Θ)3(1nn∑s=1ss+1(|Φ″(a1)|q+|Φ″(a1+a2−Θ)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2))1q] |
Corollary 3.8. Taking u→0+ in Theorem 3.3, we have
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)(p(α+1)p(α+1)+1)1p[(Θ−a1)2α+3(1nn∑s=1ss+1(|Φ″(a2)|q+|Φ″(a1+a2−Θ)|q))1q+(a2−Θ)2α+3(1nn∑s=1ss+1(|Φ″(a1)|q+|Φ″(a1+a2−Θ)|q))1q] |
Corollary 3.9. Under assumptions of Theorem 3.3 with Θ=a1+a22, we have the following inequality
|2(1−α)B(α)Φ(a1+a22)−[ABIαa2{Φ(a1+a22)}+ABa1Iα(a1+a22){Φ(a1+a22)}]+(a2−a1)α2αB(α)Γ(α)[Φ(a1)+Φ(a2)]|≤1(α+1)B(α)Γ(α)(p(α+1)p(α+1)+1)1p[(a2−a12)α+2(1nn∑s=1ss+1(|Φ″(a2)|q+|Φ″(a1+a22)|q)−2u‖a2−a1‖σ(σ+1)(σ+2))1q+(a2−a12)α+2(1nn∑s=1ss+1(|Φ″(a1)|q+|Φ″(a1+a22)|q)−2u‖a2−Θ‖σ(σ+1)(σ+2))1q] |
Theorem 3.4. Let Φ:[a1,a2]→R be a differentiable function on (a1,a2) with 0≤a1<a2. If |Φ″|q is higher order strongly n-polynomial convex on [a1,a2] for q≥1, then for all Θ∈[a1,a2] the following inequality for fractional integrals holds:
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)(α+1α+2)1−1q[(Θ−a1)α+2(|Φ″(a2)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a2−Θ)|qnn∑s=1(1−11+s−12+α+12+s+α)−u‖Θ−a1‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q+(a2−Θ)α+2(|Φ″(a1)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a2−Θ)|qnn∑s=1(1−11+s−12+α+12+s+α)−u‖a2−Θ‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q] |
Proof. Using Lemma 2.2 and the property of modulus, we have
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)[(Θ−a1)α+2∫10|1−τα+1||Φ″(τa2+(1−τ)(a1+a2−Θ))|dτ+(a2−Θ)α+2∫10|τα+1−1||Φ″(τa1+(1−τ)(a1+a2−Θ))|dτ] |
Using power mean inequality and the higher order strongly n-polynomial convexity of |Φ″|q, we get
∫10|1−τα+1||Φ″(τa2+(1−τ)(a1+a2−Θ))|dτ≤(∫10|1−τα+1|dτ)1p(∫10(1−τα+1)|Φ″(τa2+(1−τ)(a1+a2−Θ))|qdτ)1q≤(α+1α+2)1p(∫10(1−τα+1)[1nn∑s=1[1−(1−τ)s]|Φ″(a2)|q+1nn∑s=1[1−τs]|Φ″(a1+a2−Θ)|q−u(τσ(1−τ)+τ(1−τ)σ)‖Θ−a1‖σ]dτ)1q=(α+1α+2)1p(|Φ″(a2)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a2−Θ)|qnn∑s=1(1−11+s−12+α+12+s+α)−u‖Θ−a1‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q. | (3.7) |
Similarly, we have
∫10|τα+1−1||Φ″(τa1+(1−τ)(a1+a2−Θ))|dτ=(α+1α+2)1p(|Φ″(a1)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a2−Θ)|qnn∑s=1(1−11+s−12+α+12+s+α)−u‖a2−Θ‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q. | (3.8) |
By the inequalities (3.7) and (3.8), we obtain required result.
Corollary 3.10. Taking α=1 in Theorem 3.4 and using Remark 2.2, we have
|Φ′(a1+a2−Θ)[(a2−Θ)2−(Θ−a1)2]+2[(Θ−a1)Φ(a2)+(a2−Θ)Φ(a1)]−2∫a2a1Φ(Θ)dΘ|≤(23)1−1q[(Θ−a1)3(|Φ″(a2)|qnn∑s=1(23−11+s+2Γ(1+s)Γ(4+s))+|Φ″(a1+a2−Θ)|qnn∑s=1(23−11+s+13+s)−u‖Θ−a1‖σ(2(1+σ)(2+σ)−1(3+σ)(4+σ)−6Γ(1+σ)Γ(5+σ)))1q+(a2−Θ)3(|Φ″(a1)|qnn∑s=1(23−11+s+2Γ(1+s)Γ(4+s))+|Φ″(a1+a2−Θ)|qnn∑s=1(23−11+s+13+s)−u‖a2−Θ‖σ(2(1+σ)(2+σ)−1(3+σ)(4+σ)−6Γ(1+σ)Γ(5+σ)))1q]. |
Corollary 3.11. Taking u→0+ in Theorem 3.4, we have
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)(α+1α+2)1−1q[(Θ−a1)2α+3(|Φ″(a2)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a2−Θ)|qnn∑s=1(1−11+s−12+α+12+s+α))1q+(a2−Θ)2α+3(|Φ″(a1)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a2−Θ)|qnn∑s=1(1−11+s−12+α+12+s+α))1q] |
Corollary 3.12. Under assumptions of Theorem 3.4 with Θ=a1+a22, we have the following inequality
|2(1−α)B(α)Φ(a1+a22)+1B(α)Γ(α)(a2−a12)α[Φ(a1)+Φ(a2)]−[ABIαa2{Φ(a1+a22)}+ABa1Iα(a1+a22){Φ(a1+a22)}]|≤1(α+1)B(α)Γ(α)(α+1α+2)1−1q(a2−a12)α+2[(|Φ″(a2)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a22)|qnn∑s=1(1−11+s−12+α+12+s+α)−u‖a2−a12‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q+(|Φ″(a1)|qnn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+|Φ″(a1+a22)|qnn∑s=1(1−11+s−12+α+12+s+α)−u‖a2−a12‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q] |
Remark 3.1. Taking Θ=a1 or Θ=a2 in our main results, we can obtain several important special cases. We omit here their proofs and the details are left to the interested readers.
In this section, we discuss applications of our main results.
In this section, we discuss some applications of our main results to special means of positive real numbers. First of all, we recall some previously known concepts. For a1≠a2, we have
(1) The arithmetic mean: A(a1,a2)=a1+a22.
(2) The logarithmic mean: L(a1,a2)=a2−a1ln(a2)−ln(a1).
(3) The generalized logarithmic mean: Lnn(a1,a2)=[a2n+1−a1n+1(a2−a1)(n+1)]1n,n∈Z∖{−1,0}.
Proposition 4.1. Suppose all the assumptions of Theorem 3.1 are satisfied, then
(1)
|Am(a1,a2)−Lmm(a1,a2)|≤a2−a14(pp+1)1p[{n∑s=1ss+1(ma2(m−1)q+mA(n−1)q(a1,a2))}1q+{n∑s=1ss+1(ma1(m−1)q+mA(m−1)q(a1,a2))}1q], |
(2)
|L−1(a1,a2)−A−1(a1,a2)|≤a2−a14(pp+1)1p[{n∑s=1ss+1(a2−2q+A−2q(a1,a2))}1q+{n∑s=1ss+1(a1−2q+A−2q(a1,a2))}1q]. |
Proof. The proof is direct consequence of Theorem 3.1, by setting x=a1+a22,α=1,μ=0 and Φ(x)=xm, and Φ(x)=1x, respectively.
Proposition 4.2. Suppose all the assumptions of Theorem 3.2 are satisfied, then
(1)
|Am(a1,a2)−Lmm(a1,a2)|≤a2−a14(12)1p[ma2(m−1)qnn∑s=1(ss+1−12+1(s+1)(s+2))+mA(m−1)q(a1,a2)nn∑s=1(ss+1−12+1s+2)]1q+[ma1(m−1)qnn∑s=1(ss+1−12+1(s+1)(s+2))+mA(m−1)q(a1,a2)nn∑s=1(ss+1−12+1s+2)]1q, |
(2)
|L−1(a1,a2)−A−1(a1,a2)|≤a2−a14(12)1p[a2−2qnn∑s=1(ss+1−12+1(s+1)(s+2))+A−2q(a1,a2)nn∑s=1(ss+1−12+1s+2)]1q+[a1−2qnn∑s=1(ss+1−12+1(s+1)(s+2))+A−2q(a1,a2)nn∑s=1(ss+1−12+1s+2)]1q. |
Proof. The proof is direct consequence of Theorem 3.2, by taking x=a1+a22,α=1,μ=0 and Φ(x)=xm, and Φ(x)=1x, respectively.
In this last section, we discuss applications regarding bounded functions in absolute value of the results obtained from our main results. We suppose that the following conditions are satisfied:
|Φ′|≤ϝ1and|Φ″|≤ϝ2. |
Applying the above conditions, we have the following results.
Corollary 4.1. Under the assumptions of Theorem 3.1, the following inequality holds:
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)(αpαp+1)1p[(Θ−a1)α+1(2ϝq1nn∑s=1ss+1−2u‖Θ−a1‖σ(σ+1)(σ+2))1q+(a2−Θ)α+1(2ϝq1nn∑s=1ss+1−2u‖a2−Θ‖σ(σ+1)(σ+2))1q]. |
Corollary 4.2. Under the assumptions of Theorem 3.2, the following inequality holds:
|ABIαa2{Φ(a1+a2−Θ)}+ABa1Iατ{Φ(a1+a2−Θ)}−Φ(a1+a2−Θ)B(α)Γ(α)[(Θ−a1)α+(a2−Θ)α+2(1−α)Γ(α)]|≤1B(α)Γ(α)(αα+1)1−1q[(Θ−a1)α+1[ϝq1nn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+ϝq1nn∑s=1(ss+1−1α+1+1α+s+1)−u‖Θ−a1‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ))]1q+(a2−Θ)α+1[ϝq1nn∑s=1(ss+1−1α+1+Γ(1+s)Γ(1+α)Γ(2+s+α))+ϝq1nn∑s=1(ss+1−1α+1+1α+s+1)−u‖a2−Θ‖σ(1(σ+1)(σ+2)−1(1+α+σ)(2+α+σ)−Γ(2+α)Γ(1+σ)Γ(3+α+σ))]1q] |
Corollary 4.3. Under the assumptions of Theorem 3.3, the following inequality holds:
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)(p(α+1)p(α+1)+1)1p[(Θ−a1)2α+3(2ϝq2nn∑s=1ss+1−2u‖Θ−a1‖σ(σ+1)(σ+2))1q+(a2−Θ)2α+3(2ϝq2nn∑s=1ss+1−2u‖a2−Θ‖σ(σ+1)(σ+2))1q] |
Corollary 4.4. Under the assumptions of Theorem 3.4, the following inequality holds:
|Φ′(a1+a2−Θ)(α+1)B(α)Γ(α)[(a2−Θ)α+1−(Θ−a1)α+1]−[ABIαa2{Φ(a1+a2−Θ)}+ABa1Iαa1+a2−Θ{Φ(a1+a2−Θ)}]+2(1−α)B(α)Φ(a1+a2−Θ)+1B(α)Γ(α)[(Θ−a1)αΦ(a2)+(a2−Θ)αΦ(a1)]|≤1(α+1)B(α)Γ(α)(α+1α+2)1−1q[(Θ−a1)2α+3(ϝq2nn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+ϝq2nn∑s=1(1−11+s−12+α+12+s+α)−u‖Θ−a1‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q+(a2−Θ)2α+3(ϝq2nn∑s=1(1−11+s−12+α+Γ(1+s)Γ(2+α)Γ(3+s+α))+ϝq2nn∑s=1(1−11+s−12+α+12+s+α)−u‖a2−Θ‖σ(2(1+σ)(2+σ)−1(2+α+σ)(3+α+σ)−Γ(3+α)Γ(1+σ)Γ(4+α+σ)))1q] |
In 2015 Caputo and Fabrizio suggested a new operator with fractional order, this derivative is based on the exponential kernel. Earlier this year 2016 Atangana and Baleanu developed another version which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Both operators show some properties of filter. However, the Atangana and Baleanu version has in addition to this, all properties of fractional derivative. This shown effectiveness and advantages of the Atangana-Baleanu integral operators. Inspired by this great fact that own Atangana-Baleanu integral operators, we found two new fractional integral identities involving Atangana-Baleanu fractional integrals. Applying these identities as auxiliary results, we derived new fractional counterparts of classical inequalities essentially using first and second order differentiable higher order strongly n-polynomial convex functions. We have discussed several important special cases from our main results. The efficiency of our main results is demonstrated via special means and differentiable functions of first and second order that are in absolute value bounded. We will derive as future works several new fractional integral inequalities using Chebyshev, Markov, Young and Minkowski inequalities. Since the class of higher order strongly n-polynomial convex functions have large applications in many mathematical areas, they can be applied to obtain several results in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities and may stimulate further research in different areas of pure and applied sciences. Studies relating convexity, partial convexity, and preinvex functions (as contractive operators) may have useful applications in complex interdisciplinary studies, such as maximizing the likelihood from multiple linear regressions involving Gauss-Laplace distribution. For more details, see [49,50,51,52,53,54,55,56]. We hope that our ideas and techniques of this paper will inspire interested readers working in this field.
This research was funded by Dirección de Investigación from Pontificia Universidad Católica del Ecuador in the research project entitled, "Some integrals inequalities and generalized convexity" (Algunas desigualdades integrales para funciones con algún tipo de convexidad generalizada y aplicaciones).
The authors declare that they have no competing interests.
[1] |
P. O. Mohammed, T. Abdeljawad, M. A. Alqudah, F. Jarad, New discrete inequalities of Hermite-Hadamard type for convex functions, Adv. Differ. Equ., 122 (2021). https://doi.org/10.1186/s13662-021-03290-3 doi: 10.1186/s13662-021-03290-3
![]() |
[2] |
P. O. Mohammed, M. Z. Sarikaya, D. Baleanu, On the generalized Hermite-Hadamard inequalities via the tempered fractional integrals, Symmetry, 12 (2020), 595. https://doi.org/10.3390/sym12040595 doi: 10.3390/sym12040595
![]() |
[3] |
P. O. Mohammed, T. Abdeljawad, D. Baleanu, A. Kashuri, F. Hamasalh, P. Agarwal, New fractional inequalities of Hermite-Hadamard type involving the incomplete gamma functions, J. Inequal. Appl., 263 (2020). https://doi.org/10.1186/s13660-020-02538-y doi: 10.1186/s13660-020-02538-y
![]() |
[4] |
G. Rahman, K. S. Nisar, T. Abdeljawad, S. Ullah, Certain fractional proportional integral inequalities via convex functions, Mathematics, 8 (2020), 1–11. https://doi.org/10.3390/math8020222 doi: 10.3390/math8020222
![]() |
[5] |
G. Rahman, T. Abdeljawad, F. Jarad, K. S. Nisar, Bounds of generalized proportional fractional integrals in general form via convex functions and their applications, Mathematics, 8 (2020). https://doi.org/10.3390/math8010113 doi: 10.3390/math8010113
![]() |
[6] |
M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
![]() |
[7] |
H. Budak, P. Agarwal, New generalized midpoint type inequalities for fractional integral, Miskolc Math. Notes, 20 (2019), 781–793. https://doi.org/10.18514/MMN.2019.2525 doi: 10.18514/MMN.2019.2525
![]() |
[8] | Y. M. Chu, M. U. Awan, M. Z. Javed, A. G. Khan, Bounds for the remainder in Simpson's inequality via n-polynomial convex functions of higher order using Katugampola fractional integrals, J. Math., 2020 (2020). |
[9] | Y. M. Chu, M. U. Awan, S. Talib, M. A. Noor, K. I. Noor, Generalizations of Hermite-Hadamard like inequalities involving χk-Hilfer fractional integrals, Adv. Differ. Equ., 594 (2020). |
[10] | A. Kashuri, M. U. Awan, M. A. Noor, Fractional integral identity, estimation of its bounds and some applications to trapezoidal quadrature rule, Filomat, 34 (2020), 2629–2641. |
[11] | K. Liu, J. R. Wang, D. O'Regan, On the Hermite-Hadamard type inequality for ψ-Riemann-Liouville fractional integrals via convex functions, J. Inequal. Appl., 27 (2019). |
[12] | H. K. Onalan, A. O. Akdemir, M. A. Ardic, D. Baleanu, On new general versions of Hermite-Hadamard type integral inequalities via fractional integral operators with Mittag-Leffler kernel, J. Inequal. Appl., 186 (2021). |
[13] | S. Talib, M. U. Awan, Estimations of upper bounds for n-th order differentiable functions involving χ-Riemann-Liouville integrals via γ-preinvex functions, Math. Prob. Eng., 2021 (2021), 6882882. |
[14] | S. Wu, M. U. Awan, M. V. Mihai, M. A. Noor, S. Talib, Estimates of upper bound for a k-th order differentiable functions involving Riemann-Liouville integrals via higher order strongly h-preinvex functions, J. Inequal. Appl., 227 (2019). |
[15] | S. Wu, M. U. Awan, M. U. Ullah, S. Talib, A. Kashuri, Some integral inequalities for n-polynomial ζ-preinvex functions, J. Funct. Spaces, 2021 (2021), 6697729. |
[16] | Y. Zhang, T. S. Du, H. Wang, Some new k-fractional integral inequalities containing multiple parameters via generalized (s,m)-preinvexity, Ital. J. Pure Appl. Mat., 40 (2018), 510–527. |
[17] | C. J. Huang, G. Rahman, K. S. Nisar, A. Ghaffar, F. Qi, Some inequalities of the Hermite-Hadamard type for k-fractional conformable integrals, Aust. J. Math. Anal. Appl., 16 (2019). |
[18] |
G. Rahman, T. Abdeljawad, F. Jarad, A. Khan, K. S. Nisar, Certain inequalities via generalized proportional Hadamard fractional integral operators, Adv. Differ. Equ., 454 (2019). https://doi.org/10.1186/s13662-019-2381-0 doi: 10.1186/s13662-019-2381-0
![]() |
[19] |
G. Rahman, A. Khan, T. Abdeljawad, K. S. Nisar, The Minkowski inequalities via generalized proportional fractional integral operators, Adv. Differ. Equ., 287 (2019). https://doi.org/10.1186/s13662-019-2229-7 doi: 10.1186/s13662-019-2229-7
![]() |
[20] |
G. Rahman, K. S. Nisar, F. Qi, Some new inequalities of the Gruss type for conformable fractional integrals, AIMS Math., 3 (2018), 575–583. https://doi.org/10.3934/Math.2018.4.575 doi: 10.3934/Math.2018.4.575
![]() |
[21] |
K. S. Nisar, G. Rahman, K. Mehrez, Chebyshev type inequalities via generalized fractional conformable integrals, J. Inequal. Appl., 245 (2019). https://doi.org/10.1186/s13660-019-2197-1 doi: 10.1186/s13660-019-2197-1
![]() |
[22] |
K. S. Nisar, A. Tassaddiq, G. Rahman, A. Khan, Some inequalities via fractional conformable integral operators, J. Inequal. Appl., 217 (2019). https://doi.org/10.1186/s13660-019-2170-z doi: 10.1186/s13660-019-2170-z
![]() |
[23] |
K. A. Abro, I. Khan, K. S. Nisar, Novel technique of Atangana and Baleanu for heat dissipation in transmission line of electrical circuit, Chaos Soliton. Fract., 129 (2019), 40–45. https://doi.org/10.1016/j.chaos.2019.08.001 doi: 10.1016/j.chaos.2019.08.001
![]() |
[24] |
K. S. Nisar, F. Qi, G. Rahman, S. Mubeen, M. Arshad, Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function, J. Inequal. Appl., 135 (2018). https://doi.org/10.1186/s13660-018-1717-8 doi: 10.1186/s13660-018-1717-8
![]() |
[25] |
G. Rahman, Z. Ullah, A. Khan, E. Set, K. S. Nisar, Certain Chebyshev-type inequalities involving fractional conformable integral operators, Mathematics, 7 (2019). https://doi.org/10.3390/math7040364 doi: 10.3390/math7040364
![]() |
[26] |
T. Toplu, M. Kadakal, İ. İșcan, On n-polynomial convexity and some related inequalities, AIMS Math., 5 (2020), 1304–1318. https://doi.org/10.3934/math.2020089 doi: 10.3934/math.2020089
![]() |
[27] | B. T. Polyak, Existence theorems and convergence of minimizing sequences for extremal problems with constraints, Dokl. Akad. Nauk SSSR, 166 (1966), 287–290. |
[28] |
D. Baleanu, H. Khan, H. Jafari, R. A. Khan, M. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Differ. Equ., 318 (2015). https://doi.org/10.1186/s13662-015-0651-z doi: 10.1186/s13662-015-0651-z
![]() |
[29] |
H. Khan, W. Chen, H. Sun, Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space, Math. Method. Appl. Sci., 2018. https://doi.org/10.1002/mma.4835. doi: 10.1002/mma.4835
![]() |
[30] |
A. Khan, H. Khan, J. F. G. Aguilar, T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos Soliton. Fract., 127 (2019), 422–427. https://doi.org/10.1016/j.chaos.2019.07.026 doi: 10.1016/j.chaos.2019.07.026
![]() |
[31] | A. Ekinci, M. E. Özdemir, E. Set, New integral inequalities of Ostrowski type for quasi-convex functions with applications, Turk. J. Sci., 5 (2020), 290–304. |
[32] |
A. O. Akdemir, S. I. Butt, M. Nadeem, M. A. Ragusa, New general variants of Chebyshev type inequalities via generalized fractional integral operators, Mathematics, 9 (2021). https://doi.org/10.3390/math9020122 doi: 10.3390/math9020122
![]() |
[33] | S. Kızıl, M. A. Ardiç, Inequalities for strongly convex functions via Atangana-Baleanu integral operators, Turk. J. Sci., 6 (2021), 96–109. |
[34] | A. Ekinci, M. E. Özdemir, Some new integral inequalities via Riemann-Liouville integral operators, Appl. Comput. Math., 3 (2019), 288–295. |
[35] | S. I. Butt, M. Nadeem, G. Farid, On Caputo fractional derivatives via exponential s-convex functions, Turk. J. Sci., 5 (2020), 140–146. |
[36] | K. S. Miller, B. Ross, An introduction to the fractional Calculus and fractional differential equations, Wiley, New York, NY, USA, 1993. |
[37] |
A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Soliton. Fract., 89 (2016), 447–454. https://doi.org/10.1016/j.chaos.2016.02.012 doi: 10.1016/j.chaos.2016.02.012
![]() |
[38] | A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. |
[39] |
T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107. https://doi.org/10.22436/jnsa.010.03.20 doi: 10.22436/jnsa.010.03.20
![]() |
[40] |
H. Ahmad, M. Tariq, S. K. Sahoo, S. Askar, A. E. Abouelregal, K. M. Khedher, Refinements of Ostrowski type integral inequalities involving Atangana-Baleanu fractional integral operator, Symmetry, 13 (2021). https://doi.org/10.3390/sym13112059 doi: 10.3390/sym13112059
![]() |
[41] | A. A. Lupaș, A. Cǎtaș, Fuzzy differential subordination of the Atangana-Baleanu fractional integral, Symmetry, 13 (2021). |
[42] | A. A. Lupaș, A. Cǎtaș, An application of the principle of differential subordination to analytic functions involving Atangana-Baleanu fractional integral of Bessel functions, Symmetry, 13 (2021). |
[43] |
A. Khan, D. Khan, I. Khan, M. Taj, I. Ullah, A. M. Aldawsari, et al., MHD flow and heat transfer in sodium alginate fluid with thermal radiation and porosity effects: Fractional model of Atangana-Baleanu derivative of non-local and non-singular kernel, Symmetry, 11 (2019). https://doi.org/10.3390/sym11101295 doi: 10.3390/sym11101295
![]() |
[44] | E. Uçar, S. Uçar, F. Evirgen, N. Özdemir, A fractional SAIDR model in the frame of Atangana-Baleanu derivative, Fractal Fract., 5 (2021). |
[45] |
C. N. Angstmann, B. A. Jacobs, B. I. Henry, Z. Xu, Intrinsic discontinuities in solutions of evolution equations involving fractional Caputo-Fabrizio and Atangana-Baleanu operators, Mathematics, 8 (2020). https://doi.org/10.3390/math8112023 doi: 10.3390/math8112023
![]() |
[46] |
D. Baleanu, R. Darzi, B. Agheli, Existence results for Langevin equation involving Atangana-Baleanu fractional operators, Mathematics, 8 (2020). https://doi.org/10.3390/math8030408 doi: 10.3390/math8030408
![]() |
[47] |
J. B. Liu, S. I. Butt, J. Nasir, A. Aslam, A. Fahad, J. Soontharanon, Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator, AIMS Math., 7 (2022), 2123–2141. https://doi.org/10.3934/math.2022121 doi: 10.3934/math.2022121
![]() |
[48] |
A. I. K. Butt, W. Ahmad, M. Rafiq, D. Baleanu, Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic, Alex. Eng. J., 61 (2022), 7007–7027. https://doi.org/10.1016/j.aej.2021.12.042 doi: 10.1016/j.aej.2021.12.042
![]() |
[49] |
X. S. Zhou, C. X. Huang, H. J. Hu, L. Liu, Inequality estimates for the boundedness of multilinear singular and fractional integral operators, J. Inequal. Appl., 303 (2013). https://doi.org/10.1186/1029-242X-2013-303 doi: 10.1186/1029-242X-2013-303
![]() |
[50] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., 204 (2006). |
[51] |
D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics, 7 (2019). https://doi.org/10.3390/math7090830 doi: 10.3390/math7090830
![]() |
[52] | H. M. Srivastava, P. W. Karlsson, Multiple gaussian hypergeometric series, Halsted Press (Ellis Horwood Limited, Chichester): Chichester, UK, 1985. |
[53] | N. S. Barnett, P. Cerone, S. S. Dragomir, J. Roumeliotis, Some inequalities for the dispersion of a random variable whose pdf is defined on a finite interval, J. Inequal. Pure Appl. Math., 2 (2001), 1–18. |
[54] | N. S. Barnett, S. S. Dragomir, Some elementary inequalities for the expectation and variance of a random variable whose pdf is defined on a finite interval, RGMIA Res. Rep. Colloq., 2 (1999), 1–7. |
[55] |
P. Cerone, S. S. Dragomir, On some inequalities for the expectation and variance, Korean J. Comput. Appl. Math., 2 (2000), 357–380. https://doi.org/10.1007/BF02941972 doi: 10.1007/BF02941972
![]() |
[56] | J. E. Pečarič, F. Proschan, Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press: New York, NY, USA, 1991. |
1. | Miguel Vivas–Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Muhammad Aslam Noor, Silvestru Sever Dragomir, Bullen-Mercer type inequalities with applications in numerical analysis, 2024, 96, 11100168, 15, 10.1016/j.aej.2024.03.093 |