Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called (h1,h2)-Godunova-Levin preinvex functions, which is extensions of h-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.
Citation: Waqar Afzal, Najla M. Aloraini, Mujahid Abbas, Jong-Suk Ro, Abdullah A. Zaagan. Hermite-Hadamard, Fejér and trapezoid type inequalities using Godunova-Levin Preinvex functions via Bhunia's order and with applications to quadrature formula and random variable[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 3422-3447. doi: 10.3934/mbe.2024151
[1] | Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Taghreed M. Jawa . Fuzzy-interval inequalities for generalized preinvex fuzzy interval valued functions. Mathematical Biosciences and Engineering, 2022, 19(1): 812-835. doi: 10.3934/mbe.2022037 |
[2] | Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Khadijah M. Abualnaja . Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions. Mathematical Biosciences and Engineering, 2021, 18(5): 6552-6580. doi: 10.3934/mbe.2021325 |
[3] | Péter Kórus, Juan Eduardo Nápoles Valdés, Bahtiyar Bayraktar . Weighted Hermite–Hadamard integral inequalities for general convex functions. Mathematical Biosciences and Engineering, 2023, 20(11): 19929-19940. doi: 10.3934/mbe.2023882 |
[4] | Yuwen Du, Bin Nie, Jianqiang Du, Xuepeng Zheng, Haike Jin, Yuchao Zhang . New research for detecting complex associations between variables with randomness. Mathematical Biosciences and Engineering, 2024, 21(1): 1356-1393. doi: 10.3934/mbe.2024059 |
[5] | Yefu Zheng, Jun Xu, Hongzhang Chen . TOPSIS-based entropy measure for intuitionistic trapezoidal fuzzy sets and application to multi-attribute decision making. Mathematical Biosciences and Engineering, 2020, 17(5): 5604-5617. doi: 10.3934/mbe.2020301 |
[6] | Xin Chen, Tengda Li, Will Cao . Optimizing cancer therapy for individuals based on tumor-immune-drug system interaction. Mathematical Biosciences and Engineering, 2023, 20(10): 17589-17607. doi: 10.3934/mbe.2023781 |
[7] | Haiping Ren, Laijun Luo . A novel distance of intuitionistic trapezoidal fuzzy numbers and its-based prospect theory algorithm in multi-attribute decision making model. Mathematical Biosciences and Engineering, 2020, 17(4): 2905-2922. doi: 10.3934/mbe.2020163 |
[8] | Aftab Ahmed, Javed I. Siddique . The effect of magnetic field on flow induced-deformation in absorbing porous tissues. Mathematical Biosciences and Engineering, 2019, 16(2): 603-618. doi: 10.3934/mbe.2019029 |
[9] | Jing Cai, Jianfeng Yang, Yongjin Zhang . Reliability analysis of s-out-of-k multicomponent stress-strength system with dependent strength elements based on copula function. Mathematical Biosciences and Engineering, 2023, 20(5): 9470-9488. doi: 10.3934/mbe.2023416 |
[10] | Veronika Schleper . A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences and Engineering, 2015, 12(2): 393-413. doi: 10.3934/mbe.2015.12.393 |
Convex and preinvex functions are two different concepts. Specifically, preinvex functions are generalizations of convex functions. We created some intriguing examples to demonstrate how these classes differ from one another. We showed that Godunova-Levin invex sets are always convex but the converse is not always true. In this note, we present a new class of preinvex functions called (h1,h2)-Godunova-Levin preinvex functions, which is extensions of h-Godunova-Levin preinvex functions defined by Adem Kilicman. By using these notions, we initially developed Hermite-Hadamard and Fejér type results. Next, we used trapezoid type results to connect our inequality to the well-known numerical quadrature trapezoidal type formula for finding error bounds by limiting to standard order relations. Additionally, we use the probability density function to relate trapezoid type results for random variable error bounds. In addition to these developed results, several non-trivial examples have been provided as proofs.
Mathematical sciences rely heavily on convexity and it contributes to many fields such as optimization theory, economics, engineering, variational inequalities, management science and Riemannian manifolds. Convex sets and functions simplify complex problems, making them amenable to efficient computational solutions. A wide spectrum of scientific and engineering disciplines continues to benefit from concepts derived from convex analysis. Convexity is a powerful mathematical concept that can be used to simplify complicated mathematical problems and offer a theoretical framework for the creation of effective algorithms in a variety of domains. Complex systems behaviour can be deeply understood through integral inequalities that are derived from convexity concepts. These inequalities give mathematics rigour. Their ability to model, comprehend, and forecast a wide range of natural phenomena makes them indispensable instruments for engineers and physicists. Our understanding of the physical world will likely be further enhanced by the discovery of new applications and connections made possible by this field of study. To sum up, Jensen's work and later advancements in convex analysis have clarified the utility of convex functions, which is essential to understanding optimisation problems. It offers both useful techniques and theoretical underpinnings for identifying the best answers in a variety of applications. Convexity is still a major topic in mathematics, with research and applications being done in many different areas.
Approximation theory and probability distributions use generalised convexity concepts to approximate non-convex functions with convex functions. Numerous computational and numerical methods can benefit from this approximation. To summarise, integral inequalities and generalised convexity are closely related fields of study that share a mathematical framework for establishing and analysing these inequalities. The significance of comprehending the interaction between generalised convexity and integral inequalities in theoretical and practical contexts is emphasised by the applications of these ideas in a variety of fields, such as physics, functional analysis, and optimisation. Literature contains a variety of inequality types. The most crucial factor in optimization problems is Hermite-Hadamard or often called double inequality. In this context, we consider the well-known inequality owing to Hadamard and Hermite independently for convex functions; see Ref [1].
V(gg+fg2)≤1fg−gg∫fgggV(ν)dν≤V(gg)+V(fg)2. | (1.1) |
In addition to its mathematical relevance and its widespread application in a variety of domains involving different classes of generalized convexity, researchers are also investigating how to extend it to function spaces; see Refs. [2,3,4,5]. In mathematical optimization and related areas, invex functions have become important extensions of convex functions. Initially, in [6], authors introduced invex functions, that generalized classical convex mappings and discuss some of its interesting properties. In [7], Ben and Mond combined work and introduced modified form of invex sets and preinvex functions, an extension and generalization of classical convex mappings. The differentiable preinvex mappings in this class of invexity are invex, which is one of its distinguish features, but not the converse. Even though preinvex functions aren't convex, they have some lovely properties that convex functions don't.; see Ref [8]. Based on Almutairi's [9] formulation, a function V is called to be h-Godunova-Levin (GL) preinvex on interval [gg,gg+ς(fg,gg)] iff it satisfies the following double inequality
h(12)2V(2gg+ς(fg,gg)2)≤1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)dϱ≤[V(gg)+V(fg)]∫10dηoh(ηo). | (1.2) |
Based on this result, several authors extend this result in various ways, many of them using various types of preinvex functions; see Refs [10,11,12].
Interval-valued analysis allows one to deal with uncertainties and errors in a number of computational tasks effectively. This method ensures that results are based on uncertainties in input data by representing numerical values as intervals, making it particularly useful for applications that require accurate predictions and reliable results. By representing numerical values as intervals, it provides a realistic and conservative approach to computations. Through Moore's [13] contributions, interval analysis has developed a wide range of applications that span many fields, including math, computer science, engineering, and natural science; see Refs [14,15]. As a result of precise results in a variety of disciplines, mathematicians are motivated to extend integral inequalities to interval-valued mappings.
Initially, authors in [16] used h-convex mappings to link Jensen type and Hadamard type results in the setup of set-valued functions. By combining the concepts of set-valued analysis and h-convex mappings authors in [17] developed three well-known inequalities that shed light on the characteristics and behaviour of stochastic processes within a probability space. In [18], authors used the notion of preinvex functions to create double inequality for set-valued mappings. In [19] authors utilize the notion of preinvex functions on coordinates and developed various results of the double inequality on rectangular plane. Zhou, Saleem, Nazeer, Shah [20] developed an improved form of the double inequalities by using pre-invex exponential type functions via fractional integrals in the context of set-valued mappings. Khan, Catas, Aloraini, Soliman [21] used up-down preinvex mappings in a fuzzy setup to get Fejér and Hermite-type findings. Using the concept of (h1,h2)-preinvex mappings, Aslam, Khalida, Saima [22] created a number of Hermite-Hadamard type results related to special functions using power mean integral inequalities. Employing the concept of harmonical (h1,h2)-Godunova-Levin functions through centre and radius interval order relation, the authors in [23] produced Hermite-Hadamard and Jensen type results, which expand upon a number of earlier discoveries. Using local fractional integrals, Sun [24] created a various new form of double inequalities for h-preinvex functions with applications. For generalized preinvex mappings, authors in [25] developed various novel variants of double inequalities with some interesting properties using the notion of (s,m,φ) type functions. Using partial order relations, Ali et al. [26] developed different new variants of Hermite-Hadamard type results based on Godunova-Levin preinvex mappings. By combining fractional operators and generalized preinvex mappings, Tariq et al. [27] developed various new Hermite-Hadamard and Fejér type results. Sitho et al. [28] used the idea of quantum integrals to demonstrate midpoint and trapezoidal inequalities for differentiable preinvex functions. Latif, Kashuri, Hussain, Delayer [29] investigated Trapezium-type inequalities for h-preinvex functions, as well as their applications to special means. Delavar [30] used fractional integrals to find new bounds for Hermite-Hadamard's trapezoid and mid-point type inequalities. Stojiljković et al. [31] developed some new bounds for Hermite-Hadamard type inequalities involving various types of convex functions using fractional operators. Afzal, Eldin, Nazeer, Galal [32] created several novel Hermite-Hadamard type results by employing the harmonical Godunova-levin function in a stochastic sense with centre and radius order. Tariq, Ahmad, Budak, Sahoo, Sitthiwirattham [33] conducted a thorough analysis using generalized preinvex functions of Hermite-Hadamard type inequalities. Afzal, Botmart [34] used the notion of h-Godunova-Levin stochastic process and developed some new bounds of Hermite-Hadamard and Jensen type inclusions. Kalsoom, Latif, Idrees, Arif, Salleh [35] created Hermite-Hadamard type inequalities for generalized strongly preinvex functions using the idea of quantum calculus. Duo, Zhou [36] created some new bounds by using fractional double integral inclusion relations having exponential kernels via interval-valued coordinated convex mappings. Furthermore, comparable outcomes applying a variety of alternative fractional operators that we refer to [37,38,39,40].
This work is novel and noteworthy since it introduces a more generalized class, referred to as (h1,h2)-Godunova-Levin preinvex functions that unify different previously reported findings by employing different choices of bifunction ς. Since convexity and preinvexity are two different concepts, and preinvexity enjoys more nice properties than classical convex mappings, a more generalized form of inequalities is deduced with this class. Furthermore, this is the first time in literature that we have identified error bounds for quadrature type formula via this class of generalized convexity furthermore we also discuss some applications for random variables within context of error bounds that also generalize different results. The majority of literature is based on partial order or pseudo order relationships which have significant flaws in some of the inequalities results since we are not able to compare two intervals. This order relationship offers the advantage of conveniently comparing intervals and, more importantly, the endpoints of interval difference is much smaller, so a more precise result can be obtained. Recently, various authors utilized Bhunias Samanata order relation to formulate various results using different classes of convexities; see Refs. [41,42]. Stojiljković, Mirkov, Radenović [43] created a number of novel tensorial trapezoid-type inequalities for convex functions of self-ddjoint operators in Hilbert spaces. Liu, Shi, Ye, Zhao [44] employed the idea of harmonically convex functions to establish new bounds for Hermite-Hadamard type inequalities by using centre and radius orders. Regarding other recent advancements employing distinct categories of convex mappings under centre and radius order, please see [45,46,47].
The literature related to developed inequalities and specifically these articles; [9,25,41] is leading us to define a new class of preinvexity for the first time and utilizing these notions, we are developing various novel variants of the famous double and Trapezoid type inequalities and their relation to Fejér's work. The arrangement of the article is designed as: following the preliminary work in Sect. 2, we present a new class of preinvexity and talk about some of its intriguing properties in Sect. 3. The main results of this paper are presented in Sect. 4, where we developed different forms of famous double type inequalities, and in Sect. 5, where we created modified Hermite-Hadamard-Fejér type results. Section 6, focuses on error bounds of numerical integration with applications to random variable via trapezoidal type inequality. Section 7, closes with a summary of some final thoughts and suggestions for additional study.
In this section, we discuss some current definitions and results that may provide support for the primary conclusions stated in the study. Furthermore, certain ideas are used in papers without being defined; see Ref. [9].
Definition 2.1 [9] Suppose Q be a subset of R, then it is called to be invex with respect to to the bifunction ς(⋅,⋅):Q×Q→Rn, if
gg+ηoς(fg,gg)∈Q |
for all gg,fg∈Q and ηo∈[0,1].
Example 2.1. Suppose Q=[−4,−3]∪[−2,3] is called to be invex with respect to ς(⋅,⋅) and mappings is defined as:
ς(ρ1,η1)={ρ1−η1if−2≤ρ1≤3,−1≤η1≤3ρ1−η1if−4≤ρ1≤−3,−4≤η1≤−3;−4−η1if−2≤ρ1≤3,−4≤η1≤−2−2−η1if−4≤ρ1≤−3,−2≤η1≤3. |
In that situation, Q is definitely invex with respect to ς(⋅,⋅), but it is clearly not a convex set.
.
Definition 2.2. [50] Suppose Q is a invex with respect to the ς(⋅,⋅). A function V:Q→R is called to be preinvex with respect to ς(⋅,⋅) if
V(gg+ηoς(fg,gg))≤ηoV(fg)+(1−ηo) V(gg) |
for all gg,fg∈Q and ηo∈[0,1].
Definition 2.3. [50] Suppose Q is a invex with respect to the ς(⋅,⋅). A function V:Q→R is called to be GL preinvex with respect to ς if
V(gg+ηoς(fg,gg))≤V(fg)ηo+V(gg)(1−ηo) |
for all gg,fg∈Q and ηo∈(0,1).
Definition 2.4. [50] Suppose Q is a invex with respect to the ς(⋅,⋅). A function V:Q→R is called to be h-preinvex with respect to ς if
V(gg+ηoς(fg,gg))≤h(ηo)V(fg)+h(1−ηo) V(gg) |
for all gg,fg∈Q and ηo∈(0,1).
Definition 2.5. [50] Suppose Q is a invex with respect to the ς(⋅,⋅). A Function V:Q→R is called to be h-GL preinvex with respect to ς if
V(gg+ηoς(fg,gg))≤V(fg)h(ηo)+V(gg)h(1−ηo) |
for all gg,fg∈Q and ηo∈(0,1).
Definition 2.6. [50] Suppose Q is a invex with respect to ς(⋅,⋅). If for all gg,fg∈Q and ηo∈[0,1],
ς(fg,fg+ηo ς(gg,fg))=−ηoς(gg,fg) | (2.1) |
and
ς(gg,fg+ηo ς(gg,fg))=(1−ηo)ς(gg,fg). | (2.2) |
for all gg,fg∈Q and ηo1,ηηo2∈[0,1], and this is said to be Condition C, if one has
ς(fg+ηηo2 ς(gg,fg),fg+ηo1ς(gg,fg))=(ηηo2−ηo1)ς(gg,fg). |
As we proceed through the article, we will cover a few basic information regarding interval analysis.
[⋄]=[⋄_,¯⋄](⋄_≤ν≤¯⋄;ν∈R),[ς]=[ς_,¯ς](ς_≤ν≤¯ς;ν∈R),[⋄]+[ς]=[⋄_,¯⋄]+[ς_,¯ς]=[⋄_+ς_,¯⋄+¯ς] |
and
Λ⋄=Λ[⋄_,¯⋄]={[Λ⋄_,Λ¯⋄], if Λ>0;{0}, if Λ=0;[Λ¯⋄,Λ⋄_], if Λ<0, |
where Λ∈R.
Let RI be the pack of all intervals and R+I be the collection of all positive intervals of set of real number R. As a next step, we define how we calculate the relation we use throughout the article. It is called midpoint and radii of interval order relation.
More precisely ⋄ can be represented as follows:
⋄=⟨⋄c,⋄r⟩=⟨¯⋄+⋄_2,¯⋄−⋄_2⟩. |
Accordingly, we can describe the CR order relation for intervals in this manner:
Definition 2.7. [45] The Bhunia and Samanta interval order relation for ⋄=[⋄_,¯⋄]=⟨⋄c,⋄r⟩ and ς=[ς_,¯ς]=⟨ςc,ςr⟩∈RI is defined as:
⋄⪯crς⟺{⋄c<ςc,if⋄c≠ςc;⋄r≤ςr,if⋄c=ςc. |
For the intervals ⋄,ς∈RI, then this relation hold ⋄⪯crς or ς⪯cr⋄.
Definition 2.8. [46] Let V:[gg,fg] be an I.V.F where V=[V_,¯V], then V is Riemann integrable (IR) on [gg,fg] iff V_ and ¯V are (IR) on [gg,fg], that is,
(IR)∫fgggV(ϱ)dϱ=[(R)∫fgggV_(ϱ)dϱ,(R)∫fggg¯V(ϱ)dϱ]. |
The pack of all I.V.F.S for Riemann integrable on [gg,fg] is denoted by IR([gg,fg]).
Theorem 2.1. [47] Let V,ηo:[gg,fg] be an I.V.F.S defined as V=[V_,¯V] and ηo=[ηo_,¯ηo]. If V(ϱ)⪯crηo(ϱ) ∀ ϱ∈[gg,fg], then
∫fgggV(ϱ)dϱ⪯CR∫fgggηo(ϱ)dϱ. |
With the help of an example, we show that the preceding Theorem holds true.
Example 2.2. Let V=[ϱ,2ϱ] and ηo=[ϱ2,ϱ2+2]. Then, for ϱ∈[0,1],VC=3ϱ2,VR=ϱ2,ηoC=ϱ2+1 and ηoR=1. As a result, by utilizing the Definition 2.7, one has V(ϱ)⪯CRηo(ϱ) for ϱ∈[0,1]. Since,
∫10[ϱ,2ϱ]dϱ=[12,1] |
and
∫10[ϱ2,ϱ2+2]dϱ=[13,73]. |
From Theorem 2.1, one has
∫10V(ϱ)dϱ⪯CR∫10ηo(ϱ)dϱ. |
The purpose of this section is to introduce a new type of preinvexity called Godunova-Levin preinvex functions of the (h1,h2) type, based on total order relations, that generalizes several existing definitions.
Definition 3.1. Suppose V:[gg,fg] be an set-valued function given by V=[V_,¯V]. Let h1,h2:(0,1)→(0,∞) where h1,h2≠0, then V is called to be CR−(h1,h2)−GL-preinvex with respect to ς if
V(gg+ηoς(fg,gg))⪯CRV(fg)H(ηo,1−ηo)+V(gg)H(1−ηo,ηo), |
for all gg,fg∈Q and ηo∈(0,1).
Remark 3.1. Choosing h1(ηo)=1ηo,h2(ηo)=1, in Definition 3.1, the CR−(h1,h2)−GL-preinvex function reduces to the CR-preinvex function.
V(gg+ηoς(fg,gg))⪯CRηoV(fg)+(1−ηo) V(gg). |
Remark 3.2. Choosing h1(ηo)=1ηso,h2(ηo)=1, in Definition 3.1, the CR−(h1,h2)−GL-preinvex function reduces to the CR-s-preinvex function.
V(gg+ηoς(fg,gg))⪯CRηsoV(fg)+(1−ηo)s V(gg). |
Remark 3.3. Choosing h1(ηo)=ηo,h2(ηo)=1, in Definition 3.1, the CR−(h1,h2)−GL-preinvex function reduces to the CR-GL-preinvex function.
V(gg+ηoς(fg,gg))⪯CRV(fg)ηo+ V(gg)(1−ηo). |
Remark 3.4. Choosing h1(ηo)=1ηo(1−ηo),h2(ηo)=1, in Definition 3.1, the CR−(h1,h2)−GL-preinvex function reduces to the tgs CR preinvex function [52].
V(gg+ηoς(fg,gg))⪯CRηo(1−ηo)[V(fg)+V(gg)]. |
Remark 3.5. Choosing ς(fg,gg)=fg−gg and h1(ηo)=1ηo,h2(ηo)=1, in Definition 3.1, the CR−(h1,h2)−GL-preinvex function reduces to the CR-convex function [52].
V(ηofg+(1−ηo)gg)⪯CRηoV(fg)+(1−ηo) V(gg). |
Remark 3.6. Choosing V_=¯V, in Definition 3.1, the CR−(h1,h2)−GL-preinvex function reduces to the h-GL-preinvex function [48].
V(gg+ηoς(fg,gg))≤V(fg)h(ηo)+V(gg)h(1−ηo). |
Proposition 3.1. Let V:[gg,fg]→RI be an set-valued function given by V=[V_,¯V]=⟨VC,VR⟩. If VC and VR are (h1,h2)-GL-preinvex functions, then V is a CR-(h1,h2)-GL-preinvex mapping.
Proof. Since VC and VR are (h1,h2)-GL-preinvex functions, and ∀ ηo∈(0,1), one has
VC(gg+ηoς(fg,gg))≤VC(fg)H(ηo,1−ηo)+VC(gg)H(1−ηo,ηo) |
and
VR(gg+ηoς(fg,gg))≤VR(fg)H(ηo,1−ηo)+VR(gg)H(1−ηo,ηo). |
If VC(gg+ηoς(fg,gg))≠VC(fg)H(ηo,1−ηo)+VC(gg)H(1−ηo,ηo), then
VC(gg+ηoς(fg,gg))<VC(fg)H(ηo,1−ηo)+VC(gg)H(1−ηo,ηo). |
This implies
VC(gg+ηoς(fg,gg))⪯CRVC(fg)H(ηo,1−ηo)+VC(gg)H(1−ηo,ηo). |
Otherwise, VR(gg+ηoς(fg,gg))≤VR(fg)H(ηo,1−ηo)+VR(gg)H(1−ηo,ηo) this implies
VR(gg+ηoς(fg,gg))⪯CRVR(fg)H(ηo,1−ηo)+VR(gg)H(1−ηo,ηo). |
From Definition 3.1, we have
V(gg+ηoς(fg,gg))⪯CRV(fg)H(ηo,1−ηo)+V(gg)H(1−ηo,ηo) |
This demonstrate that, if VC and VR are (h1,h2)-GL-preinvex functions, then V is a CR-(h1,h2)-GL-preinvex function.
As part of this section, we present several new Hermite-Hadamard and Fejér type inequalities for Godunova-Levin-preinvex functions of the (h1,h2) type.
Theorem 4.1. Let V:[gg,gg+ς(fg,gg)]→RI be an set-valued function defined as V(ϱ)=[V_(ϱ),¯V(ϱ)]. If V:[gg,gg+ς(fg,gg)]→R is a CR-(h1,h2)-GL-preinvex mapping and satisfies the Condition C, then the following relation holds:
[H(12,12)]2V(2gg+ς(fg,gg)2)⪯CR1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)dϱ⪯CR[V(gg)+V(fg)]∫10dηoH(ηo,1−ηo). |
Proof. By definition of CR-(h1,h2)-GL-preinvex function, one has
V(2ν1+ς(ν2,ν1)2)⪯CR1[H(12,12)][V(ν1)+V(ν2)]. |
Choosing ν1=gg+ηoς(fg,gg) and ν2=gg+(1−ηo)ς(fg,gg), we have
V(gg+ηoς(fg,gg)+12ς(gg+(1−ηo)ς(fg,gg),gg+ηoς(fg,gg)))⪯CR1[H(12,12)][V(gg+ηoς(fg,gg))+V(gg+(1−ηo)ς(fg,gg))]. |
This implies
[H(12,12)]V(2gg+ς(fg,gg)2)⪯CR[V(gg+ηoς(fg,gg))+V(gg+(1−ηo)ς(fg,gg))]. | (4.1) |
Integrating aforementioned inequality (4.1), we obtain
[H(12,12)]V(2gg+ς(fg,gg)2)⪯CR[∫10V(gg+ηoς(fg,gg))dηo+∫10V(gg+(1−ηo)ς(fg,gg))dηo]=∫10(V_(gg+ηoς(fg,gg))+V_(gg+(1−ηo)ς(fg,gg)))dηo,∫10(¯V(gg+ηoς(fg,gg))+¯V(gg+(1−ηo)ς(fg,gg)))dηo=2ς(fg,gg)∫gg+ς(fg,gg)ggV_(ϱ)dϱ,2ς(fg,gg)∫gg+ς(fg,gg)gg¯V(ϱ)dϱ=2ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)dϱ. |
From the previous developments, we can infer that
[H(12,12)]2V(2gg+ς(fg,gg)2)⪯CR1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)dϱ. | (4.2) |
From Definition 3.1, we have
V(gg+ηoς(fg,gg))⪯CRV(fg)H(ηo,1−ηo)+V(gg)H(1−ηo,ηo). |
Integrating the above result, we get
∫10V(gg+ηoς(fg,gg))dηo⪯CRV(fg)∫10dηoH(ηo,1−ηo)+V(gg)∫10dηoH(1−ηo,ηo). |
This implies
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)dϱ⪯CR[V(gg)+V(fg)]∫10dηoH(ηo,1−ηo). | (4.3) |
By combining (4.2) and (4.3), we get required result.
Note: Based on our newly developed results, several previously published results have been unified.
Remark 4.1. ● Choosing h1(ηo)=h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.1 generates outcomes for CR-h-GL functions [41].
● Choosing h1(ηo)=1h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.1 generates outcomes for CR-h-convex functions [51].
● Choosing h1(ηo)=1h1(ηo),h2(ηo)=1h2(ηo) and ς(fg,gg)=fg−gg, then Theorem 4.1 generates outcomes for CR-(h1,h2)-convex functions [49].
Example 4.1. Let V(ϱ)=[1−ϱ12,(9−3ϱ12)], ς(fg,gg)=fg−gg, fg=2 and gg=0, then for h1(ηo)=1ηo,h2(ηo)=1, we have
[H(12,12)]2V(2gg+ς(fg,gg)2)≈[0,5.999],1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)dϱ≈[0.057,6.171],[V(gg)+V(fg)]∫10dηoH(ηo,1−ηo)≈[0.585,13.757]. |
As a result, Theorem 4.1 is validated as accurate.
[0,5.999]⪯CR[0.057,6.171]⪯CR[0.585,13.757]. |
Theorem 4.2. Let V,Y:[gg,gg+ς(fg,gg)]→RI be an set-valued functions, which are given by Y(ϱ)=[Y_(ϱ),¯Y(ϱ)] and V(ϱ)=[V_(ϱ),¯V(ϱ)]. If V,Y:[gg,gg+ς(fg,gg)]→R are CR-(h1,h2)-GL-preinvex functions, then the following double inequality applies:
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)Y(ϱ)dϱ⪯CRM(gg,fg)∫10dηoH2(ηo,1−ηo)+N(gg,fg)∫10dηoH(1−ηo,1−ηo)H(ηo,ηo). | (4.4) |
where
M(gg,fg)=V(gg)Y(gg)+V(fg)Y(fg) |
and
N(gg,fg)=V(gg)Y(fg)+V(fg)Y(gg). |
Proof. Since V,Y are CR-(h1,h2)-GL-preinvex functions, we have
V(gg+yς(fg,gg))⪯CRV(fg)H(ηo,1−ηo)+V(gg)H(1−ηo,ηo) |
and
Y(gg+yς(fg,gg))⪯CRY(fg)H(ηo,1−ηo)+Y(gg)H(1−ηo,ηo). |
The product of the two aforementioned results gives us
V(gg+yς(fg,gg))Y(gg+yς(fg,gg))⪯CR[V(fg)H(ηo,1−ηo)+V(gg)H(1−ηo,ηo)][Y(fg)H(ηo,1−ηo)+Y(gg)H(1−ηo,ηo)]=[V(fg)Y(fg)]H2(ηo,1−ηo)+[V(gg)Y(gg)]H2(1−ηo,ηo)+[V(fg)Y(gg)]+[V(gg)Y(fg)]H(1−ηo,1−ηo)H(ηo,ηo). | (4.5) |
For integrating (4.5), we have
∫10V(gg+yς(fg,gg))Y(gg+yς(fg,gg))dηo⪯CR[V(fg)Y(fg)]∫10dηoH2(ηo,1−ηo)+[V(gg)Y(gg)]∫10dηoH2(1−ηo,ηo)+[V(fg)Y(gg)+V(gg)Y(fg)]∫10dηoH(1−ηo,1−ηo)H(ηo,ηo). |
From Definition 2.8, we obtain
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)Y(ϱ)dϱ⪯CR[V(gg)Y(gg)+V(fg)Y(fg)]∫10dηoH2(1−ηo,ηo)+[V(gg)Y(fg)+V(fg)Y(gg)]∫10dηoH(1−ηo,1−ηo)H(ηo,ηo)=M(gg,fg)∫10dηoH2(1−ηo,ηo)+N(gg,fg)∫10dηoH(1−ηo,1−ηo)H(ηo,ηo). |
Remark 4.2. Choosing h1(ηo)=h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.2 generates outcomes for CR-h-GL functions [41].
Remark 4.3. Choosing h1(ηo)=1h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.2 generates outcomes for CR-h-convex functions [51].
1fg−gg∫fgggV(ϱ)Y(ϱ)dϱ⪯CRM(gg,fg)∫10h(ηo)2dηo+N(gg,fg)∫10h(1−ηo)h(ηo)dηo. |
Remark 4.4. Choosing h1(ηo)=1h1(ηo),h2(ηo)=1h2(ηo) and ς(fg,gg)=fg−gg, then Theorem 4.2 generates outcomes for CR-(h1,h2)-convex functions [49].
1fg−gg∫fgggV(ϱ)Y(ϱ)dϱ⪯CRM(gg,fg)∫10H2(ηo,1−ηo)dηo+N(gg,fg)∫10H(1−ηo,1−ηo)H(ηo,ηo)dηo. |
Remark 4.5. Choosing h1(ηo)=1y,h2(ηo)=1, then Theorem 4.2 generates outcomes for CR-preinvex functions, i.e.,
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)Y(ϱ)dϱ⪯CRM(gg,fg)3+N(gg,fg)6. |
Remark 4.6. Choosing h1(ηo)=1y,h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.2 generates outcomes for CR-convex functions, i.e.,
1fg−gg∫fgggV(ϱ)Y(ϱ)dϱ⪯CRM(gg,fg)3+N(gg,fg)6. |
Example 4.2. Let V(ϱ)=[2−ϱ12,(6−3ϱ12)],Y(ϱ)=[eϱ−ϱ,eϱ+ϱ], ς(fg,gg)=fg−gg,gg=0 and fg=2. Then, for h1(ηo)=1y,h2(ηo)=1, we have
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)Y(ϱ)dϱ≈[1.95,10.9] |
and
M(gg,fg)∫10dηoH2(ηo,1−ηo)+N(gg,fg)∫10dηoH(1−ηo,1−ηo)H(ηo,ηo)dηo≈[4.32,15.96]. |
Thus, we have
[1.95,10.9]⪯CR[4.32,15.96]. |
Theorem's 4.2 validity is therefore confirmed.
Theorem 4.3. Following the same hypothesis as Theorem 4.2, the following relationship holds:
[H(12,12)]22V(2gg+ς(fg,gg)2)Y(2gg+ς(fg,gg)2)⪯CR1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)Y(ϱ)dϱ+M(gg,fg)∫10dηoH(1−ηo,1−ηo)H(ηo,ηo)+N(gg,fg)∫10dηoH2(ηo,1−ηo). |
Proof. The proof is completed by taking into account Definition 3.1 and using the same technique as [An et al. [53], Theorem 5].
Remark 4.7. Choosing h1(ηo)=1h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.3 generates outcomes for CR-h-convex functions [51].
Remark 4.8. Choosing h1(ηo)=1h1(ηo),h2(ηo)=1h2(ηo) and ς(fg,gg)=fg−gg, then Theorem 4.3 generates outcomes for CR-(h1,h2)-convex functions [49].
Remark 4.9. Choosing h1(ηo)=h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 4.3 generates outcomes for CR-h-GL functions [41].
Example 4.3. Suppose V(ϱ)=[−ϱ2,2ϱ2+1],Y(ϱ)=[−ϱ,ϱ],ς(fg,gg)=fg−gg,gg=1 and fg=3. Then, for h1(ηo)=1y,h2(ηo)=14, we have
[H(12,12)]22V(gg+12ς(fg,gg))Y(gg+12ς(fg,gg))≈[−1.031,1.031] |
and
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)Y(ϱ)dϱ+M(gg,fg)∫10dηoH(ηo,ηo)H(1−ηo,1−ηo)+N(gg,fg)∫10dηoH2(ηo,1−ηo)≈[−132.25,45]. |
Thus, we have
[−1.031,1.031]⪯CR[−132.25,45]. |
Theorem's 4.3 validity is therefore confirmed.
Theorem 5.1. Let V:[gg,gg+ς(fg,gg)]→RI be an set-valued function is defined as V(ϱ)=[V_(ϱ),¯V(ϱ)] for all ϱ∈[gg,fg]. If V:[gg,gg+ς(fg,gg)]→R is an CR-(h1,h2)-GL-preinvex and W:[gg,gg+ς(fg,gg)]→R is symmetric with respect to gg+12ς(fg,gg), then the following outcome holds:
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ⪯CR[V(gg)+V(fg)]∫10W(gg+yς(fg,gg))dηoH(ηo,1−ηo). |
Proof. As V is an CR-(h1,h2)-GL-preinvex function and W is symmetric function, we have
V(gg+yς(fg,gg))W(gg+yς(fg,gg))⪯CR[V(fg)H(ηo,1−ηo)+V(gg)H(1−ηo,ηo)]W(gg+yς(fg,gg)) |
and
V(gg+(1−ηo)ς(fg,gg))W(gg+(1−ηo)ς(fg,gg))⪯CR[V(gg)H(ηo,1−ηo)+V(fg)H(1−ηo,ηo)]W(gg+(1−ηo)ς(fg,gg)). |
Including the two aforementioned results and then integrating, we have
∫10V(gg+yς(fg,gg))W(gg+yς(fg,gg))dηo | (5.1) |
+∫10V(gg+(1−ηo)ς(fg,gg))W(gg+(1−ηo)ς(fg,gg))dηo⪯CR∫10[V(gg)((W(gg+yς(fg,gg))H(1−ηo,ηo)+W(gg+(1−ηo)ς(fg,gg))H(ηo,1−ηo))+V(fg)((W(gg+yς(fg,gg))H(ηo,1−ηo)+W(gg+(1−ηo)ς(fg,gg)))H(1−ηo,ηo))]dηo=2V(gg)∫10W(gg+(1−ηo)ς(fg,gg))H(ηo,1−ηo)dηo+2V(fg)∫10W(gg+yς(fg,gg))H(ηo,1−ηo)dηo=2[V(gg)+V(fg)]∫10W(gg+yς(fg,gg))H(ηo,1−ηo)dηo. | (5.2) |
Since
∫10V(gg+yς(fg,gg))W(gg+yς(fg,gg))dηo | (5.3) |
+∫10V(gg+(1−ηo)ς(fg,gg))W(gg+(1−ηo)ς(fg,gg))dηo=2ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ, | (5.4) |
We achieve the desired outcome by accounting results (5.1) and (5.3).
Remark 5.1. If h1(ηo)=1h(ηo),h2(ηo)=1 with V_=¯V, then Theorem 5.1 generates outcomes for h-GL-preinvex functions, i.e.,
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ≤[V(gg)+V(fg)]∫10W(gg+yς(fg,gg))h(ηo)dηo. |
Remark 5.2. If h1(ηo)=1y,h2(ηo)=1, then Theorem 5.1 generates outcomes for CR-preinvex functions, i.e.,
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ⪯CR[V(gg)+V(fg)]∫10yW(gg+yς(fg,gg))dηo. |
Remark 5.3. If h1(ηo)=1h(ηo),h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 5.1 generates outcomes for CR−h−GL functions, i.e.,
1fg−gg∫fgggV(ϱ)W(ϱ)dϱ⪯CR[V(gg)+V(fg)]∫10W((1−ηo)gg+yfg)h(ηo)dηo. |
Example 5.1. Suppose V(ϱ)=[3−√ϱ,8−√ϱ)],ς(fg,gg)=fg−gg,gg=0 and fg=2. Then, for h1(ηo)=1y,h2(ηo)=1, W(ϱ)=ϱ for ϱ∈[0,1] and W(ϱ)=−ϱ+3 for ϱ∈[1,2], one has
1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ=12∫20V(ϱ)W(ϱ)dϱ=12∫10[(3−ϱ12)ϱ,ϱ(8−ϱ12)]dϱ+12∫21[(3−ϱ12)(−ϱ+3),(−ϱ+3)(8−4ϱ12)]dϱ≈[1.9029,3.6117] |
and
[V(gg)+V(fg)]∫10W(gg+yς(fg,gg))H(ηo,1−ηo)dηo=([3,8]+[3−212,(8−4√2)])∫10yW(2y)dηo=[6−212,(16−4√2)](∫1202y2dt+∫112y(−2y+3)dηo)≈[2.8661,6.4646]. |
Thus, we have
[1.9029,3.6117]⪯CR[2.8661,6.4646]. |
Theorem's 5.1 validity is therefore confirmed.
Theorem 5.2. Following the same hypothesis as Theorem 5.1, the following relationship holds true
V(2gg+ς(fg,gg)2)⪯CR2[H(12,12)]∫gg+ς(fg,gg)ggW(ϱ)dϱ∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ. |
Proof. As V is an CR-(h1,h2)-GL-preinvex function, one has
V(2gg+ς(fg,gg)2)⪯CR1[H(12,12)][V(gg+yς(fg,gg))+V(gg+(1−ηo)ς(fg,gg))]. |
Multiplying aforementioned inequality by W(gg+yς(fg,gg))=W(gg+(1−ηo)ς(fg,gg)) and integrating, we have
V(2gg+ς(fg,gg)2)∫10W(gg+yς(fg,gg))dηo⪯CR1[H(12,12)][∫10V(gg+yς(fg,gg))W(gg+yς(fg,gg))dηo+∫10V(gg+(1−ηo)ς(fg,gg))W(gg+(1−ηo)ς(fg,gg))dηo]. | (5.5) |
Since
∫10V(gg+yς(fg,gg))W(gg+yς(fg,gg))dηo=∫10V(gg+(1−ηo)ς(fg,gg))W(gg+(1−ηo)ς(fg,gg))dηo=1ς(fg,gg)∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ | (5.6) |
and
∫10W(gg+yς(fg,gg))dηo=1ς(fg,gg)∫gg+ς(fg,gg)ggW(ϱ)dϱ. | (5.7) |
Using (5.6) and (5.7) in (5.5), we have
V(2gg+ς(fg,gg)2)⪯CR2[H(12,12)]∫gg+ς(fg,gg)ggW(ϱ)dϱ∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ. |
Remark 5.4. If V_=¯V, then Theorem 5.2 generates outcomes for (h1,h2)-GL-preinvex function, i.e.,
V(2gg+ς(fg,gg)2)≤2[H(12,12)]∫gg+ς(fg,gg)ggW(ϱ)dϱ∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ. |
Remark 5.5. If h1(ηo)=1y,h2(ηo)=1, then Theorem 5.2 generates outcomes for CR-preinvex functions, i.e.,
V(2gg+ς(fg,gg)2)⪯CR1∫gg+ς(fg,gg)ggW(ϱ)dϱ∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ. |
Remark 5.6. If ς(fg,gg)=fg−gg, then Theorem 5.2 generates outcomes for CR-(h1,h2)-GL function, i.e.,
V(gg+fg2)⪯CR2[H(12,12)]∫fgggW(ϱ)dϱ∫fgggV(ϱ)W(ϱ)dϱ. |
Remark 5.7. If h1(ηo)=1y,h2(ηo)=1 and ς(fg,gg)=fg−gg, then Theorem 5.2 generates outcomes for CR-convex functions, i.e.,
V(gg+fg2)⪯CR1∫fgggW(ϱ)dϱ∫fgggV(ϱ)W(ϱ)dϱ. |
Example 5.2. Following the same hypothesis as Example 5.1, we have
V(gg+12ς(fg,gg))=V(1)=[2,7] |
and
2H(12,12)∫gg+ς(fg,gg)ggW(ϱ)dϱ∫gg+ς(fg,gg)ggV(ϱ)W(ϱ)dϱ=1∫20W(ϱ)dϱ∫20[3−√ϱ,(8−4√ϱ)]W(ϱ)dϱ≈[3.80588,7.22354]. |
Thus, we have
[2,7]⪯CR[3.80588,7.22354]. |
Theorem's 5.2 validity is therefore confirmed.
This section aims to develop several applications of the numerical quadrature rule, specifically the trapezoid type rule, using the standard order relation (≤) via generalised convexity defined in [53].
Theorem 6.1. Consider Y:I⊆R→R be a differentiable function on I∘,gg,fg∈I∘ with gg<fg and V:[gg,fg]→R+be a differentiable function symmetric to gg+fg2. If |Y′| is an (h1,h2)-convex function on [gg,fg], then
|Y(gg)+Y(fg)2∫fgggV(ν)dν−∫fgggY(ν)V(ν)dν|≤(fg−gg)(|Y′(gg)|+|Y′(fg)|)∫fg+gg2gg∫10V(ν)[H(ηo,1−ηo)+H(1−ηo,ηo)]dηodν. |
where
ZV(ηo)={2∫12yV(sgg+(1−s)fg)ds0≤y≤12−2∫y12V(sgg+(1−s)fg)ds12≤y≤1. |
Proof. From the definition of ZV(ηo) and (h1,h2)- convexity of |Y′| we have
|Y(gg)+Y(fg)2∫fgggV(ν)dν−∫fgggY(ν)V(ν)dν|=(fg−gg)22|∫10ZV(ηo)Y′(ygg+(1−ηo)fg)dy|≤(fg−gg)22{∫120|ZV(ηo)||Y′(ygg+(1−ηo)fg)|dy+∫112|ZV(ηo)||Y′(ygg+(1−ηo)fg)|dy}=(fg−gg)22{∫120ZV(ηo)|Y′(ygg+(1−ηo)fg)|dy−∫112ZV(ηo)|Y′(ygg+(1−ηo)fg)|dy}≤(fg−gg)22{2∫120∫12yV(sgg+(1−s)fg)(H(ηo,1−ηo)|Y′(gg)|+h1(1−ηo)h2(ηo)|Y′(fg)|)dsdηo+2∫112∫y12V(sgg+(1−s)fg)(H(ηo,1−ηo)|Y′(gg)|+h1(1−ηo)h2(ηo)|Y′(fg)|)dsdηo}. |
Modify the integration order,
|Y(gg)+Y(fg)2∫fgggV(ν)dν−∫fgggY(ν)V(ν)dν|≤(fg−gg)2{∫120∫s0V(sgg+(1−s)fg)(H(ηo,1−ηo)|Y′(gg)|+H(1−ηo,ηo)|Y′(fg)|)dηods+∫112∫1sV(sgg+(1−s)fg)(H(ηo,1−ηo)|Y′(gg)|+H(1−ηo,ηo)|Y′(fg)|)dηods}. |
Using the variable change ν=sgg+(1−s)fg, one has
|Y(gg)+Y(fg)2∫fgggV(ν)dν−∫fgggY(ν)V(ν)dν|≤(fg−gg){∫fgfg+gg2∫fg−νfg−gg0V(ν)(H(ηo,1−ηo)|Y′(gg)|+h1(1−ηo)h2(ηo)|Y′(fg)|)dηodν+∫fg+gg2gg∫1fg−νfg−ggV(ν)(H(ηo,1−ηo)|Y′(gg)|+H(1−ηo,ηo)|Y′(fg)|)dηodν}. | (6.1) |
As V is symmetric to fg+gg2, then one has
∫fgfg+gg2∫fg−νfg−gg0V(ν)(H(ηo,1−ηo)|Y′(gg)|+H(1−ηo,ηo)|Y′(fg)|)dηodν=∫fg+gg2gg∫ν−ggfg−gg0V(ν)(H(ηo,1−ηo)|Y′(gg)|+H(1−ηo,ηo)|Y′(fg)|)dηodν. | (6.2) |
Replacing (6.2) in (6.1) it follows that
|Y(gg)+Y(fg)2∫fgggV(ν)dν−∫fgggY(ν)V(ν)dν|≤(fg−gg)(|Y′(gg)|+|Y′(fg)|)∫fg+gg2gg∫10V(ν)[H(ηo,1−ηo)+H(1−ηo,ηo)]dηodν. | (6.3) |
Consider p be a partition of [gg,fg], i.e., p:gg=v0<v1<⋯<vn−1<vn=fg, of this quadrature formula
∫fgggY(ν)V(ν)dν=T(Y,V,p)+S(Y,V,p), |
where
T(Y,V,p)=n−1∑i=0Y(ei)+Y(ei+1)2∫ei+1eiV(ν)dν, |
is called to be trapezoidal formula. Consider a subinterval [ei,ei+1] while using Theorem 6.1. This gives the following as:
|Y(ei)+Y(ei+1)2∫ei+1eiV(ν)dν−∫ei+1eiY(ν)V(ν)dν|≤(ei+1−ei)[|Y′(ei)|+|Y′(ei+1)|]∫ei+1ei+ei+12∫ei+1−νei+1−ei0V(ν)[H(ηo,1−ηo)+H(1−ηo,ηo)]dηodν, | (6.4) |
Using the inequality (6.4) and the triangular inequality, we obtain
∣T(Y,V,p)−∫fgggY(ν)V(ν)dν∣=|n−1∑i=0[Y(ei)+Y(ei+1)2∫ei+1eiV(ν)dν−∫ei+1eiY(ν)V(ν)dν]|≤n−1∑i=0|Y(ei)+Y(ei+1)2∫ei+1eiV(ν)dν−∫ei+1eiY(ν)V(ν)dν|≤n−1∑i=0(ei+1−ei)[|Y′(ei)|+|Y′(ei+1)|]∫ei+1ei+ei+12∫ei+1−νei+1−ei0V(ν)×[H(ηo,1−ηo)+H(1−ηo,ηo)]dηodν. |
This provides us with the error bound:
|S(Y,V,p)|≤n−1∑i=0(ei+1−ei)[|Y′(ei)|+|Y′(ei+1)|]×∫ei+1ei+ei+12∫ei+1−νei+1−ei0V(ν)[H(ηo,1−ηo)+H(1−ηo,ηo)]dηodν. |
Remark 6.1. If h1(ηo)=yk,h2(ηo)=1 with k=1 in (6.4), then we reiterate the disparity revealed in [54].
|S(Y,p)|≤18n−1∑i=0[|Y′(ei)|+|Y′(ei+1)|](ei+1−ei)2. |
Applications to Random Variable
Consider a probability density function. V:[gg,fg]→R+ with 0<gg<fg, then
∫fgggV(ν)dν=1, |
which is symmetric to fg+gg2 and let u be a moment where u∈R then, we have
Eu(X)=∫fgggνuV(ν)dν, |
is finite. From Theorem (6.1) and the fact that for any gg≤ν≤fg+gg2 we have 0≤ν−ggfg−gg≤12, the following result holds.
|Y(gg)+Y(fg)2∫fgggV(ν)dν−∫fgggY(ν)V(ν)dν|≤(fg−gg)(|Y′(gg)|+|Y′(fg)|)×∫fg+gg2gg∫120V(ν)[H(ηo,1−ηo)+H(1−ηo,ηo)]dηodν=(fg−gg)2(|Y′(gg)|+|Y′(fg)|)×∫120[H(ηo,1−ηo)+H(1−ηo,ηo)]dy, |
since V is symmetric and ∫fgggV(ν)dν=1, we have ∫fg+gg2ggV(ν)dν=12.
Example 6.1. If we take into account
{Y(ν)=1uνu,ν>0,u∈(−∞,0)∪(0,2]∪[3,+∞);h1(ηo)=yk,h2(ηo)=14k∈(−∞,−1)∪(−1,1];V(ν)=1. |
Since |Y′| is (h1,h2)-convex and so from Theorem 6.1 we have
|ggu+fgu2u−Eu(X)|≤u(fg−gg)2(ggu−1+fgu−1)∫120[yk4+(1−ηo)k4]dy=u(fg−gg)4(k+1)(ggu−1+fgu−1). |
As a result, the required bound is
|ggu+fgu2u−Eu(X)|≤u(fg−gg)4(k+1)(ggu−1+fgu−1), |
Remark 6.2. If u=1,h2(ηo)=1,k=1, then we can get the following known bound as follows:
|fg+gg2−E(X)|≤fg−gg4. |
In this work, Godunova-Levin type mappings via set-valued functions are used to study a variety of inequalities associated with a new class of preinvexity. To start, we define the Godunova-Levin preinvex mappings under the full-order relation and examine some of its induced properties. We generalize many previously reported results and build novel forms by using arbitrary non-negative functions and related bifunctions of Hermite, Hadamard, and Fejér-type inequalities. We also discuss some special cases of these inequalities. To further illustrate the accuracy of the obtained results, a few numerical examples are given. In the subsequent, we concentrate on numerical integration error bounds and their applications to random variables through trapezoidal type inequality, utilising standard order via generalized convexity. Further research into other kinds of convex inequalities is feasible using the idea and concepts established in this work, with potential applications to issues like differential equations with convex shapes attached and optimisation problems.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. NRF-2022R1A2C2004874). This work was supported by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Ministry of Trade, Industry & Energy(MOTIE) of the Republic of Korea (No. 20214000000280).
The authors declare there is no conflict of interest.
[1] | J. Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor, J. de Math. Pures et Appl., 9 (1892), 101–186. |
[2] |
W. Afzal, N. Aloraini, M. Abbas, J. S. Ro, A. A. Zaagan, Some Novel Kulisch-Miranker Type Inclusions for a Generalized Class of Godunova-Levin Stochastic Processes, AIMS Math., 9 (2024), 5122-–5146. https://doi.org/10.3934/math.2024249 doi: 10.3934/math.2024249
![]() |
[3] |
M. Bessenyei, The Hermite–Hadamard Inequality in Beckenbach's Setting, J. Math. Anal. Appl., 364 (2010), 366–-383. http://doi.org/10.1016/j.jmaa.2009.11.015 doi: 10.1016/j.jmaa.2009.11.015
![]() |
[4] |
W. Afzal, M. Abbas, W. Hamali, A. M. Mahnashi, M. D. Sen, Hermite-Hadamard-Type inequalities via Caputo-Fabrizio fractional integral for h-Godunova-Levin and (h1,h2)-Convex functions, Fractal Fract., 7 (2023), 687. https://doi.org/10.3390/fractalfract7090687 doi: 10.3390/fractalfract7090687
![]() |
[5] |
V. Stojiljković, Simpson Type Tensorial Norm Inequalities for Continuous Functions of Selfadjoint Operators in Hilbert Spaces, Creat. Math. Inform., 33 (2024), 105–117. https://doi.org/10.37193/CMI.2024.01.10 doi: 10.37193/CMI.2024.01.10
![]() |
[6] |
M. A. Hanson, On sufficiency of the Kun-Tucker conditions, J. Math. Anal. Appl., 90 (1981), 545–-550. https://doi.org/10.1016/0022-247X(81)90123-2 doi: 10.1016/0022-247X(81)90123-2
![]() |
[7] |
T. Weir, B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl., 136 (1988), 29–38. https://doi.org/10.1016/0022-247X(88)90113-8 doi: 10.1016/0022-247X(88)90113-8
![]() |
[8] | M. A. Noor, Hermite–Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Appl. Approx. Theory., 2 (2007), 126–13. |
[9] |
O. Almutairi, A. Kılıçman, Some Integral Inequalities for h-Godunova-Levin Preinvexity, Symmetry, 11 (2019), 1500. https://doi.org/10.3390/sym11121500 doi: 10.3390/sym11121500
![]() |
[10] |
S. R. Mohan, S. K. Neogy, On invex set and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908. https://doi.org/10.1006/jmaa.1995.1057 doi: 10.1006/jmaa.1995.1057
![]() |
[11] |
A. A. Ahmadini, W. Afzal, M. Abbas, E. S. Aly, Weighted Fejér, Hermite–Hadamard, and Trapezium-Type Inequalities for (h1,h2)–Godunova–Levin Preinvex Function with Applications and Two Open Problems, Mathematics, 12 (2024), 382. https://doi.org/10.3390/math12030382 doi: 10.3390/math12030382
![]() |
[12] | M. A. Latif, S. S. Dragomir, M. Abbas, Some Hermite–Hadamard type inequalities for fuctions whose partial derivatives in absolut value are preinvex on the coordinates, Facta Univ. Math. Inform., 28 (2013), 257–270. |
[13] | R. E. Moore, Method and Applications of Interval Analysis, Philadelphia: Society for Industrial and Applied Mathematics, 1979. |
[14] |
E. de Weerdt, Q. P. Chu, J. A. Mulder, Neural network output optimization using interval analysis, IEEE T. Neural Network, 20 (2009), 638–653. http://doi.org/10.1109/TNN.2008.2011267 doi: 10.1109/TNN.2008.2011267
![]() |
[15] |
W. Afzal, M. Abbas, S. M. Eldin, Z. A. Khan, Some well known inequalities for (h1,h2)-convex stochastic process via interval set inclusion relation, AIMS Math., 8 (2023), 19913–19932. http://dx.doi.org/10.3934/math.20231015 doi: 10.3934/math.20231015
![]() |
[16] |
D. Zhao, T. An, G. Ye, W. Liu, Khan, New Jensen and Hermite–Hadamard type inequalities for h-convex interval-valued functions, J. Ineq. Appl., 1 (2018), 1–14. http://dx.doi.org/10.1186/s13660-018-1896-3 doi: 10.1186/s13660-018-1896-3
![]() |
[17] |
W. Afzal, E. Prosviryakov, S. M. El-Deeb, Y. Almalki, Some new estimates of Hermite-Hadamard, Ostrowski and Jensen-type inclusions for h-convex stochastic process via interval-valued functions, Symmetry, 15 (2023), 831. https://doi.org/10.3390/sym15040831 doi: 10.3390/sym15040831
![]() |
[18] |
H. M. Srivastava, S. K. Sahoo, P. O. Mohammad, D. Baleanu, B. Kodamasingh, Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators, Int. J. Comput. Intell. Syst., 15 (2022), 8. https://doi.org/10.1007/s44196-021-00061-6 doi: 10.1007/s44196-021-00061-6
![]() |
[19] |
K. K. Lai, S. K. Mishra, J. Bishat, M. Hassan, Hermite–Hadamard Type Inclusions for Interval-Valued Coordinated Preinvex Functions, Symmetry, 14 (2022), 771. https://doi.org/10.3390/sym14040771. doi: 10.3390/sym14040771
![]() |
[20] |
H. Zhou, M. S. Saleem, W. Nazeer, A. F. Shah, Hermite-Hadamard Type Inequalities for Interval-Valued Exponential Type Pre-Invex Functions via Riemann-Liouville Fractional Integrals, AIMS Math., 7 (2022), 2602–2617. https://doi.org/10.3934/math.2022146. doi: 10.3934/math.2022146
![]() |
[21] |
M. B. Khan, A. Catas, N. Aloraini, M, S, Soliman, Some Certain Fuzzy Fractional Inequalities for Up and Down h-Pre-Invex via Fuzzy-Number Valued Mappings, Fractal Fract., 7 (2023), 171. https://doi.org/10.3390/fractalfract7020171 doi: 10.3390/fractalfract7020171
![]() |
[22] |
M. A. Noor, K. I. Noor, S. Rashid, Some New Classes of Preinvex Functions and Inequalities, Mathematics, 7 (2018), 29. https://doi.org/10.3390/math7010029 doi: 10.3390/math7010029
![]() |
[23] |
Saeed, W. Afzal, M. Abbas, S. Treanţă, M. De la Sen, Some new generalizations of integral inequalities for Harmonical Cr-(h1,h2)-Godunova-Levin functions and applications, Mathematics, 10 (2022), 4540. https://doi.org/10.3390/math10234540 doi: 10.3390/math10234540
![]() |
[24] |
W. Sun, Some Hermite–Hadamard Type Inequalities for Generalized h-Preinvex Function via Local Fractional Integrals and Their Applications, Adv. Diff. Equ., 2020 (2020), 426. https://doi.org/10.1186/s13662-020-02812-9 doi: 10.1186/s13662-020-02812-9
![]() |
[25] |
A. Kashuri, R. Likho, Hermite-Hadamard Type Inequalities for Generalized (s,m,φ)-Preinvex Godunova-Levin Functions, Matematičke Znanosti, 6 (2018), 63–75. https://doi.org/10.21857/m16wjc6rl9 doi: 10.21857/m16wjc6rl9
![]() |
[26] |
S. Ali, S. R. Ali, M. Vivas-Cortez, S. Mubeen, G. Rahman, K. S. Nisar, Some Fractional Integral Inequalities via h-Godunova-Levin Preinvex Function, AIMS Math., 7 (2022), 13832–13844. https://doi.org/10.3934/math.2022763 doi: 10.3934/math.2022763
![]() |
[27] |
M. Tariq, S. K. Sahoo, S. K. Ntouyas, M. O. Alsalmai, A. A. Shaikh, K. Nonlaopon, Some Hermite–Hadamard and Hermite–Hadamard–Fejér Type Fractional Inclusions Pertaining to Different Kinds of Generalized Preinvexities, Symmetry, 14 (2022), 1957. https://doi.org/10.3390/sym14101957 doi: 10.3390/sym14101957
![]() |
[28] |
S. Sitho, M. A. Ali, H. Budak, S. K. Ntouyas, J. Tariboon, Alsalmai, A. A. Shaikh, K. Nonlaopon, Trapezoid and Midpoint Type Inequalities for Preinvex Functions via Quantum Calculus, Mathematics, 9 (2021), 1666. https://doi.org/10.3390/math9141666 doi: 10.3390/math9141666
![]() |
[29] |
M. Latif, A. Kashuri, S. Hussain, R. Delayer, Trapezium-Type Inequalities for h-Preinvex Functions and Their Applications, Filomat, 36 (2022), 3393–3404. https://doi.org/10.2298/FIL2210393L doi: 10.2298/FIL2210393L
![]() |
[30] |
M. R. Delavar, New Bounds for Hermite-Hadamard's Trapezoid and Mid-Point Type Inequalities via Fractional Integrals, Miskolc Math. Notes, 20 (2019), 849. http://dx.doi.org/10.18514/MMN.2019.2796 doi: 10.18514/MMN.2019.2796
![]() |
[31] |
V. Stojiljković, R. Ramaswamy, F. Alshammari, O. A. Ashour, M. L. H. Alghazwani, S. Radenović, Hermite–Hadamard Type Inequalities Involving (k−p) Fractional Operator for Various Types of Convex Functions, Fractal Fract., 6 (2022), 376. https://doi.org/10.3390/fractalfract6070376 doi: 10.3390/fractalfract6070376
![]() |
[32] |
W. Afzal, S. M. Eldin, W. Nazeer, A. M. Galal, Some integral inequalities for Harmonical Cr-h-Godunova-Levin stochastic processes, AIMS Math., 8 (2023), 13473–13491. http://dx.doi.org/10.3934/math.2023683 doi: 10.3934/math.2023683
![]() |
[33] |
M. Tariq, H. Ahmad, H. Budak, S. K. Sahoo, T. Sitthiwirattham, A Comprehensive Analysis of Hermite–Hadamard Type Inequalities via Generalized Preinvex Functions, Axioms, 10 (2021), 328. https://doi.org/10.3390/axioms10040328 doi: 10.3390/axioms10040328
![]() |
[34] |
W. Afzal, T. Botmart, Some novel estimates of Jensen and Hermite–Hadamard inequalities for h-Godunova-Levin stochastic processes, AIMS Math., 8 (2023), 7277–7291. http://dx.doi.org/10.3934/math.2023366 doi: 10.3934/math.2023366
![]() |
[35] |
H. Kalsoom, M. A. Latif, M. Idrees, M. Arif, Z. Salleh, Quantum Hermite-Hadamard Type Inequalities for Generalized Strongly Preinvex Functions, AIMS Math., 6 (2021), 13291–13310. https://doi.org/10.3934/math.2021769 doi: 10.3934/math.2021769
![]() |
[36] |
T. Duo, T. Zhou, On the Fractional Double Integral Inclusion Relations Having Exponential Kernels via Interval-Valued Co-Ordinated Convex Mappings, Chaos Solit. Fract., 156 (2022), 111846. https://doi.org/10.1016/j.chaos.2022.111846 doi: 10.1016/j.chaos.2022.111846
![]() |
[37] |
T. Duo, Y. Peng, Hermite–Hadamard Type Inequalities for Multiplicative Riemann–Liouville Fractional Integrals, J. Comput. Appl. Math., 440 (2024), 115582. https://doi.org/10.1016/j.cam.2023.115582 doi: 10.1016/j.cam.2023.115582
![]() |
[38] |
T. Zhou, Z. Yuan, T. Du, On the Fractional Integral Inclusions Having Exponential Kernels for Interval-Valued Convex Functions, Math. Sci., 17 (2023), 107–120. https://doi.org/10.1007/s40096-021-00445-x doi: 10.1007/s40096-021-00445-x
![]() |
[39] |
T. Du, C. Luo, Z. Cao, On the Bullen-type inequalities via generalized fractional integrals and their applications, Fractals, 29 (2021), 2150188. https://doi.org/10.1186/s13660-022-02878-xx doi: 10.1186/s13660-022-02878-xx
![]() |
[40] |
X. Zhang, K. Shabbir, W. Afzal, H. Xiao, D. Lin, Hermite-hadamard and jensen-type inequalities via Riemann integral operator for a generalized class of godunova–levin functions, J. Math., 2022 (2022), 3830324. https://doi.org/10.1155/2022/3830324 doi: 10.1155/2022/3830324
![]() |
[41] |
W. Afzal, M. Abbas, J. E. Macias-Diaz, S. Treanţă, Some H-Godunova-Levin Function inequalities using center radius (Cr) order relation, Fractal Fract., 6 (2022), 518. https://doi.org/10.3390/fractalfract6090518 doi: 10.3390/fractalfract6090518
![]() |
[42] |
W. Afzal, W. Nazeer, T. Botmart, S. Treanţă, Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation, AIMS Math., 8 (2023), 1696–1712. http://dx.doi.org/10.3934/math.2023087 doi: 10.3934/math.2023087
![]() |
[43] |
V. Stojiljković, N. Mirkov, S. Radenović, Variations in the Tensorial Trapezoid Type Inequalities for Convex Functions of Self-Adjoint Operators in Hilbert Spaces, Symmetry, 16 (2024), 121. https://doi.org/10.3390/sym16010121 doi: 10.3390/sym16010121
![]() |
[44] |
W. Liu, F. Shi, G. Ye, D. Zhao, The Properties of Harmonically Cr-h-Convex Function and Its Applications, Mathematics, 10 (2022), 2089. https://doi.org/10.3390/math10122089 doi: 10.3390/math10122089
![]() |
[45] |
Y. Almalki, W. Afzal, Some new estimates of Hermite-Hadamard inequalities for harmonical Cr-h-convex functions via generalized fractional integral operator on set-valued mappings, Mathematics, 11 (2023), 4041. https://doi.org/10.3390/math11194041 doi: 10.3390/math11194041
![]() |
[46] |
M. Abbas, W. Afzal, T. Botmart, A. M. Galal, Ostrowski and Hermite-Hadamard type inequalities for h-convex stochastic processes by means of center-radius order relation, AIMS Math., 8 (2023), 16013–16030. http://dx.doi.org/10.3934/math.2023817 doi: 10.3934/math.2023817
![]() |
[47] |
W. Afzal, K. Shabbir, M. Arshad, J. K. K. Asamoah, A. M. Galal, Some novel estimates of integral inequalities for a generalized class of harmonical convex mappings by means of center-radius order relation, J. Math., 2023 (2023), 8865992. https://doi.org/10.1155/2023/8865992 doi: 10.1155/2023/8865992
![]() |
[48] |
M. A. Noor, K. I. Noor, M. U. Awan, J. Li, On Hermite-Hadamard Inequalities for h-Preinvex Functions, Filomat, 28 (2014), 1463–1474. http://doi.org/10.2298/FIL1407463N. doi: 10.2298/FIL1407463N
![]() |
[49] |
T. Saeed, W. Afzal, K. Shabbir, S. Treanţă, M. De la Sen, Some Novel Estimates of Hermite-Hadamard and Jensen Type Inequalities for (h1,h2)-convex functions pertaining to total order relation, Mathematics, 10 (2022), 4777. https://doi.org/10.3390/math10244777 doi: 10.3390/math10244777
![]() |
[50] |
F. Zafar, S. Mehommad, A. Asiri, Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals, Demonst. Math., 56 (2023), 2022–0254. https://doi.org/10.1515/dema-2022-0254 doi: 10.1515/dema-2022-0254
![]() |
[51] |
W. Afzal, K. Shabbir, T. Botmart, S. Treanţă, Some new estimates of well known inequalities for (h1,h2)-Godunova-Levin functions by means of center-radius order relation, AIMS Math., 8 (2023), 3101–3119. http://dx.doi.org/10.3934/math.2023160 doi: 10.3934/math.2023160
![]() |
[52] |
V. Stojiljković, R. Ramaswamy, O. A. A. Abdelnaby, S. Radenović, Riemann-Liouville Fractional Inclusions for Convex Functions Using Interval Valued Setting, Mathematics, 10 (2022), 3491. https://doi.org/10.3390/math10193491 doi: 10.3390/math10193491
![]() |
[53] |
Y. An, G. Ye, D. Zhao, W. Liu, Hermite-Hadamard type inequalities for interval (h1,h2)-Convex functions, Mathematics, 7 (2019), 436. http://dx.doi.org/10.3390/math7050436 doi: 10.3390/math7050436
![]() |
[54] |
S. S. Dragomir, R. P. Agarwal, Two Inequalities for Differentiable Mappings and Applications to Special Means of Real Numbers and to Trapezoidal Formula, Appl. Math. Letters, 11 (1998), 91–45. https://doi.org/10.1016/S0893-9659(98)00086-X doi: 10.1016/S0893-9659(98)00086-X
![]() |
1. | Waqar Afzal, Mujahid Abbas, Daniel Breaz, Luminiţa-Ioana Cotîrlă, Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents, 2024, 8, 2504-3110, 518, 10.3390/fractalfract8090518 | |
2. | Waqar Afzal, Mujahid Abbas, Jongsuk Ro, Khalil Hadi Hakami, Hamad Zogan, An analysis of fractional integral calculus and inequalities by means of coordinated center-radius order relations, 2024, 9, 2473-6988, 31087, 10.3934/math.20241499 | |
3. | Zareen A. Khan, Waqar Afzal, Mujahid Abbas, Jong-Suk Ro, Abdullah A. Zaagan, Some well known inequalities on two dimensional convex mappings by means of Pseudo $ \mathcal{L-R} $ interval order relations via fractional integral operators having non-singular kernel, 2024, 9, 2473-6988, 16061, 10.3934/math.2024778 | |
4. | Waqar Afzal, Daniel Breaz, Mujahid Abbas, Luminiţa-Ioana Cotîrlă, Zareen A. Khan, Eleonora Rapeanu, Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem, 2024, 12, 2227-7390, 1238, 10.3390/math12081238 | |
5. | Dawood Khan, Saad Ihsan Butt, Youngsoo Seol, Analysis of $(P,\mathrm{m})$-superquadratic function and related fractional integral inequalities with applications, 2024, 2024, 1029-242X, 10.1186/s13660-024-03218-x | |
6. | Zareen A. Khan, Waqar Afzal, Mujahid Abbas, Jongsuk Ro, Najla M. Aloraini, A novel fractional approach to finding the upper bounds of Simpson and Hermite-Hadamard-type inequalities in tensorial Hilbert spaces by using differentiable convex mappings, 2024, 9, 2473-6988, 35151, 10.3934/math.20241671 | |
7. | Davey K. Tomlinson, " The Persistence of Habit: Tantric Engagements with Dharmakīrti’s View of Yogic Perception ", 2025, 30669030, 10.57010/LVNA9592 |