Abstract
A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, -convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, -Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and -calculus are also included.
Keywords:
Fejér inequality; Hadamard fractional integral; conformable and non-conformable fractional integral; Katugampola fractional integral; Riemann–Liouville fractional integral; (p,q)-calculus MSC:
26A33; 26A51; 26D07; 26D10; 26D15
1. Introduction
The literature about convexity had been investigated and discussed by many intellectual mathematicians before the 1960s, first of all by Fenchel and Minkowski. The efforts of Moreau and Rockafellar, who started a systematic examination of this new subject, considerably expanded and initiated the literature on convex theory at the beginning of the 1960s. Convexity and its assumptions have grown into an intriguing discipline of applied and pure mathematics over the past century. A lot of researchers have offered and contributed their expertise and insights into this area by offering updated versions of certain inequalities involving convex functions. The use of the concept of convexity in applications, of which convex optimization [1] is the primary one, is widespread. This concept has a lot of applications in applied sciences, such as finance [2], signal processing [3], control systems [4], computer science [5], mathematical optimization for modeling [6,7], engineering [8], and statistics [9]. In the subject of economics [10], this concept performs a fundamental influence on duality theory and equilibrium.
The study of integral inequalities along with convex analysis offers a fascinating and stimulating area of study in the realm of mathematical perception. Due to its importance, the literature of these concepts has recently become an amazing topic of research in both historical and contemporary times. The H-H (Hermite–Hadamard)-type and Fejér-type inequalities are the most frequently employed among all inequalities. These convex function-based inequalities are crucial and basic in practical mathematics. Thus, convexity and inequalities have been recommended as an engrossing area for researchers due to their vital role and fruitful importance. Integral inequalities have remarkable uses in integral operator theory, stochastic processes, probability, numerical integration, statistics, optimization theory and information technology. For the applications, see the references [11,12,13,14,15].
Many scholars are currently intrigued by the topic of convex functions, notably one famous inequality involving convexity known as the H-H inequality, which is stated as:
The above inequality (1) was first time developed by C. Hermite [16] and explored by J. Hadamard [17] in 1893.
Fejér [18] was the first to introduce the following Fejér inequality (weighted version of H-H inequality), which is given by:
Theorem 1
([18]). Assume that is a convex function. Then, the inequality
holds, where is non-negative, integrable and symmetric to
Fractional calculus has captivated and motivated several researchers and mathematicians across a wide spectrum of practical and scientific disciplines. Fractional integrals and derivatives, which can interpolate between operators of integer order, have a long track record and are often employed in real-world applications, as can be seen in the references [19,20,21,22]. This calculus has enlarged to be a prominent field of investigation due to its utilization in the nonlinear systems (nonlocal) and modeling. Convex functions in the frame of the fractional integral operator have many real-world applications in modeling, circuit design, optimization, controller design, etc. This idea has attracted so much attention that it evolved into a fruitful subject for investigation and inspiration.
The intention and aim of this review manuscript are to offer an extensive and accurate overview of Fejér-type inequalities via multiple sorts of convexities pertaining to fractional calculus. In every part, we first set up the fundamental descriptions of fractional integral operators and different sorts of convexities, and then we present the results for Fejér-type fractional integral inequalities. We contend that compiling almost all current fractional Fejér-type inequalities in a single document will enable fresh scholars in the discipline to learn about previous work on the problem before creating new conclusions. We give outcomes without evidence but provide a comprehensive explanation for each outcome explored in this review for the reader’s advantage.
Very recently, the authors in [23] provide an amazing review of H-H type inequalities involving convexities in the frame of fractional integral operators. The paper [23] was complimented with [24] by an up-to-date review of H-H-type inequalities pertaining to quantum calculus.
The construction of this review paper is as follows. In Section 2, we introduce the reader to the basic concepts of Riemann–Liouville fractional integrals. In Section 2.2, Section 2.3, Section 2.4, Section 2.5, Section 2.6, Section 2.7, Section 2.8, Section 2.9, Section 2.10 and Section 2.11, we summarize Fejér-type fractional integral inequalities for various classes of convexities, including classical convex functions, s-convex functions, harmonically s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, -convex functions, and h-preinvex function. In Section 3, we present Fejér-type fractional integral inequalities using the -Riemann–Liouville fractional integrals; in Section 4, we present Fejér-type fractional integral inequalities via k-Riemann–Liouville fractional integral; in Section 5, we present Fejér-type fractional integral inequalities of fractional integrals with respect to another function; and in Section 6, we present Fejér-type fractional integral inequalities for exponential kernel. Fejér-type fractional integral inequalities via the Hadamard fractional integral are presented in Section 7; Fejér-type fractional integral inequalities via Raina integrals are presented in Section 8; Fejér-type fractional integral inequalities via conformable integrals are presented in Section 9; Fejér-type fractional integral inequalities are explored via non-conformable integrals in Section 10; Fejér-type fractional integral inequalities for Katugampola integral operators are included in Section 11; while Fejér-type fractional integral inequalities for invex functions are included in Section 12. Finally, Fejér-type fractional integral inequalities for -calculus are included in the last Section 13.
It is essential to recognize that the intention here is to present a deeper and more extensive assessment, and we opted to cover as many consequences as possible to reflect progress on the topic. Any proofs are omitted for this matter, and the reader is referred to the relative article accordingly.
2. Fejér-Type Fractional Integral Inequalities via Riemann–Liouville Fractional Integral
First, we add the definitions of Riemann–Liouville fractional integrals in the left and right aspects.
Definition 1
([25]). Assume that Then, Riemann–Liouville integrals in the left and right sense and are stated by
and
respectively. Here, represents the Euler Gamma function and
2.1. Fejér-Type Fractional Integral Inequalities for Convex Functions
In this subsection, we summarize Fejér-type fractional integral inequalities concerning convex functions.
Theorem 2
([26]). Let be a differentiable function on the interior of such that for with and be a continuous function. If is convex on then, for any and , the fractional inequality is given as:
Theorem 3
([26]). Let be a differentiable function on the interior such that for with and be a continuous function. If is convex on for some fixed with then, for any and the fractional inequality is given as:
Theorem 4
([26]). Let be a differentiable function on the interior such that for with and be a continuous function. If is convex on for some fixed then, for any and , the fractional inequality is given as:
Theorem 5
([27]). Let be a differentiable mapping on and with If is convex on and is continuous and symmetric to , the fractional inequality is given as:
where
Theorem 6
([27]). Suppose that all the conditions of Theorem 5 hold. Then, the fractional inequality is given as:
Theorem 7
([27]). Suppose that all the conditions of Theorem 5 hold. Additionally, we assume that is convex on Then, the fractional inequality is given as:
where
Theorem 8
([28]). Let be a differentiable mapping on with , and let be continuous on . If is convex on then the fractional inequality is given as:
Theorem 9
([28]). Let be a differentiable mapping on with and let be continuous on . If is convex on then the fractional inequality is given as:
where and
Theorem 10
([29]). Let be a convex function with and If is non-negative, integrable, and symmetric to then inequality in the frame of fractional operator is given as:
Theorem 11
([29]). Let be a differentiable mapping on and with If is a convex function on and is continuous and symmetric to then inequality in the frame of the fractional operator is given as:
Theorem 12
([30]). Assume that with and is a differentiable function on such that and is continuous. If is a convex function on then an inequality in the frame of the fractional operator is given as:
where
Theorem 13
([30]). Assume that Q is as in Theorem 12 and is continuous. If is a convex function on then an inequality in the frame of the fractional operator is given as:
where
2.2. Fejér-Type Fractional Integral Inequalities for s-Convex Functions
Definition 2
([31]). A function is said to be s-convex in the second sense if
for all and for some fixed
In the next theorems, Fejér-type fractional integral inequalities for s-convex functions are presented.
Theorem 14
([32]). Let be a differentiable mapping on with and let be continuous on . If is s-convex in the second sense on then the fractional inequality is given as:
where
and is the incomplete beta function defined as follows
Theorem 15
([32]). Let be a differentiable mapping on with and let be continuous on . If is s-convex in the second sense on then the fractional inequality be given as:
where and
Theorem 16
([33]). Let be a differentiable function on and where and is continuous and symmetric to . If is s-convex in the second sense on for some fixed then the fractional inequality is given as:
Theorem 17
([33]). Let be differentiable function on and where and be continuous and symmetric to . If is s-convex in the second sense on for some fixed then the fractional inequality is given as:
Theorem 18
([33]). Assume that the conditions of Theorem 17 hold true. Then:
- (i)
- For , we have:
- (ii)
- For and , we have:
Theorem 19
([34]). Let be a differentiable function on and , where and are continuous. If is s-convex in the second sense on for some fixed then the fractional inequality is given as:
Theorem 20
([34]). Let be differentiable function on and where and be continuous. If is s-convex in the second sense on for some fixed then the fractional inequality is given as:
Theorem 21
([34]). Let be differentiable function on and where and be continuous. If is s-convex in the second sense on for some fixed then the fractional inequality is given as:
Theorem 22
([35]). Let be a differentiable function on such that where with If is s-convex in the second sense on , then inequality in the frame of the fractional operator is given as:
Theorem 23
([35]). Let be a differentiable function on such that where with and is continuous. If is s-convex in the second sense on , then an inequality in the frame of fractional operator is given as:
2.3. Fejér-Type Fractional Integral Inequalities for Harmonically s-Convex Functions
In this subsection, some Fejér-type integral inequalities for harmonically s-convex functions are presented.
Definition 3
([36]). Assume that is a real interval. A function is harmonically s-convex if
for all and
Theorem 24
([37]). Let be a differentiable function on , the interior of such that where and If is harmonically s-convex on is continuous and harmonically symmetric with respect to then the fractional inequality is given as:
where
with and
Theorem 25
([37]). Let be a differentiable function on , the interior of such that where and If is harmonically s-convex on is continuous and harmonically symmetric with respect to then the fractional inequality is given as:
where
with and
Theorem 26
([37]). Let be a differentiable function on , the interior of such that where and If is harmonically s-convex on is continuous and harmonically symmetric with respect to then the fractional inequality is given as:
where
with and
2.4. Fejér-Type Fractional Integral Inequalities for Quasi-Convex Functions
Definition 4
([38]). A function is said to be quasi-convex on if
for all and
In the following, we present theorems including Fejér-type fractional integral inequalities for quasi-convex functions.
Theorem 27
([39]). Let be a differentiable mapping on and with and be continuous. If is quasi convex on then the fractional inequality is given as:
where
Theorem 28
([39]). Suppose that all the conditions of Theorem 27 hold. Then, the fractional inequality is given as:
where and
Theorem 29
([39]). Let be a differentiable mapping on and with and is continuous and symmetric to If is quasi convex on then the fractional inequality is given as:
where
Theorem 30
([39]). Let be a differentiable mapping on and with and be continuous and symmetric to If is quasi convex on then the fractional inequality is given as:
where
Theorem 31
([39]). Suppose that all the conditions of Theorem 30 hold. Then, the fractional inequality is given as:
- (i)
- where
- (ii)
- where and
2.5. Fejér-Type Fractional Integral Inequalities for Harmonically Convex Functions
Definition 5
([40]). Assume that is a real interval. A function is harmonically convex, if
for all and
Definition 6
([40]). A function is said to be harmonically symmetric with respect to if
holds for all
Fejér-type fractional integral inequalities for harmonically convex functions are presented now.
Theorem 32
([41]). Let be a differentiable function on , the interior of such that where and If is harmonically convex on is continuous and harmonically symmetric with respect to then the fractional inequality is given as:
where
with and
Theorem 33
([41]). Assume that the function g is as in Theorem 32. Let be a differentiable function on , the interior of such that where and If is harmonically convex on then the fractional inequality is given as:
where
with and
Theorem 34
([41]). Assume that the function g is as in Theorem 32. Let be a differentiable function on , the interior of such that where and If is harmonically convex on then the fractional inequality is given as:
where
with and and
Theorem 35
([42]). Let be differentiable mapping where with Assume also that is a continuous positive mapping symmetric to If the mapping is harmonically-convex on then the fractional inequality is given as:
with
and
Theorem 36
([42]). Let g be as in Theorem 35. Let be differentiable mapping where with and is harmonically convex on then the fractional inequality is given as:
with
and are given in Theorem 35.
Theorem 37
([43]). Assume that with and is a harmonically convex function and If is harmonically symmetric with respect to , integrable and non-negative then the fractional inequalities are given as:
with and
Theorem 38
([43]). Assume that Q is as in Theorem 37. If is harmonically convex on and is harmonically symmetric with respect to and continuous, then an inequality in the frame of the fractional operator is given as:
where
with and
2.6. Fejér-Type Fractional Integral Inequalities for Harmonically Quasi-Convex Functions
Definition 7
([44]). A function is said to be harmonically quasi-convex, if
for all and
Theorem 39
([45]). Assume that is a differentiable function on such that where and If is harmonically quasi-convex function on is continuous and harmonically symmetric with respect to then the fractional inequality is given as:
with and
Theorem 40
([45]). Assume that Q is as in Theorem 39. If is harmonically quasi-convex function on is continuous and harmonically symmetric with respect to then the fractional inequality is given as:
with and and
2.7. Fejér-Type Fractional Integral Inequalities for p-Convex Functions
Definition 8
([46]). The function Q is strongly p-convex with modulus μ if
holds for
We include in the next theorem a Fejér-type fractional integral inequality for strongly convex functions in a generalized sense.
Theorem 41
([47]). Let the strongly generalized p-convex function with magnitude and be bounded above in and Also, let be an integrable, non-negative and symmetric with respect to then
Definition 9
([48]). Let A function is said to be p-symmetric with respect to if
holds for all
Theorem 42
([48]). Assume that is a p-convex function, and with If and is non-negative, integrable and p-symmetric with respect to then the fractional inequalities are given as:
with and
with
2.8. Fejér-Type Fractional Integral Inequalities via Convexity with Respect to Strictly Monotone Function
Definition 10
([38]). Let be intervals in and be the convex function, also let be strictly monotone function. Then, Q is called convex with respect to σ if
for all provided is a convex set.
We give a Fejér-type fractional integral inequality for convex function Q with respect to a strictly monotone function
Theorem 43
([49]). Let be intervals in and be real valued functions. Let Q be convex and g be positive and symmetric about . Let be a strictly monotone function. If Q is convex with respect to then the fractional inequality is given as:
In the next theorem, we establish another version of the Fejér-type fractional integral inequality for a convex function with respect to a strictly monotone function.
Theorem 44
([49]). Under the conditions of Theorem 43, the fractional inequality is given as:
2.9. Fejér-Type Fractional Integral Inequalities for Co-Ordinated Convex Functions
This subsection includes the Fejér-type fractional integral inequalities for co-ordinated convex functions via fractional integral.
Definition 11
([50]). A function will be called co-ordinated convex on Δ, for all and if the following inequality holds:
Definition 12
([51]). Assume that The Riemann–Liouville integrals and of order with are defined by
respectively.
Theorem 45
([51]). Let be a co-ordinated convex on in with and If is non-negative, integrable and symmetric to and then for all and , we have the inequalities:
Theorem 46
([51]). Assume that the conditions of Theorem 45 are satisfied. Then, we have the inequalities:
Theorem 47
([52]). Let be co-ordinated convex on in with and If is non-negative, integrable, and symmetric to and then for all , we have the inequalities:
Theorem 48
([52]). Under the conditions of Theorem 47, we have the fractional inequalities:
2.10. Fejér-Type Fractional Integral Inequalities for -Convex Functions
Definition 13
([53]). Let be an interval containing and let be a non-negative function. Let be an interval and A function is said to be -convex, if
holds, provided that for and
Theorem 49
([53]). Let be a positive -convex function with If is a positive function, then
Theorem 50
([53]). Assume that Q is as in Theorem 49. Then, the following inequalities hold:
2.11. Fejér-Type Fractional Integral Inequalities for h-Preinvex Functions
Definition 14
([54]). Let and where X is a non-empty closed set in be continuous function. Assume that . Then, Q is said to be h-preinvex with respect to if
for all and where
Theorem 51
([55]). Let be a h-preinvex function, condition C for Φ holds, and , and be differentiable and symmetric to , then we have
Theorem 52
([55]). Let be an open invex subset with respect to and with . Suppose that is a differentiable mapping on A and is differentiable and symmetric to . If is h-preinvex on A, we have
Theorem 53
([55]). Let be an open invex subset with respect to and with . Suppose that is a differentiable mapping on A and is differentiable and symmetric to . If is h-preinvex on A and , we have
Theorem 54
([55]). Let be an open invex subset with respect to and with . Suppose that is a differentiable mapping on A and is differentiable and symmetric to . If is h-preinvex on A and , we have
3. Fejér-Type Fractional Integral Inequalities Using -Riemann–Liouville Fractional Integrals
In this section, we present Fejér-type fractional integral inequalities using -Riemann–Liouville fractional integrals.
Definition 15
([56]). The left- and right-sided -Riemann–Liouville fractional integral operator for a real-valued function Q is defined by
and
Theorem 55
([57]). Let be a convex function with and Then, is also convex and If is non-negative and integrable, then the following inequalities holds true
Theorem 56
([57]). Let be a differentiable mapping on with and Then, is also differentiable and If is convex on and is continuous, then the fractional inequality is given as:
where and
4. Fejér-Type Fractional Integral Inequalities via -Riemann–Liouville Fractional Integral
Definition 16
([58]). Let and The k-Riemann–Liouville fractional integrals and of order for a real-valued function Π are defined by
and
respectively, where is the k-Gamma function
In the following theorems, we give Fejér-type inequalities for quasi-convex functions via k-Riemann–Liouville fractional integrals.
Theorem 57
([59]). Assume that be a differentiable mapping on and with and is continuous. If is quasi-convex on then the fractional inequality is given as:
where
Theorem 58
([59]). Assume that Q is as in Theorem 57. Then, an inequality in the frame of the k-fractional operator is given as:
where
5. Fejér-Type Fractional Integral Inequalities of Fractional Integrals with Respect to Another Function
Now, we recall the definition of fractional integrals of real-valued function concerning to another function.
Definition 17
([25]). Let be an increasing and positive function on having a continuous derivative on . The left- and right-sided Riemann–Liouville fractional integrals of Q with respect to the function ψ on of order are defined, respectively, by
and
provided that the integrals exists.
In the following, we state Fejér-type fractional integral inequalities of fractional integrals with respect to another function.
Theorem 59
([60]). Let Let be an increasing and positive monotone function on having a continuous derivative on and let be non-negative, integrable. If Q is a convex function on then the fractional inequalities are given as:
where
Theorem 60
([61]). Let and be an increasing and positive function on having a continuous derivative on . Let be non-negative, integrable, and g be a convex function on the fractional inequalities are given as:
where
Theorem 61
([61]). Let be as in Theorem 60. Let be a differentiable function on and and is continuous function. If is s-convex in the second sense on for some fixed then the fractional inequality is given as:
where
Theorem 62
([61]). Let be as in Theorem 60. If is s-convex in the second sense on for some fixed and then the fractional inequality is given as:
Theorem 63
([61]). Let be as in Theorem 60. If is s-convex in the second sense on for some fixed and then the fractional inequality is given as:
Fejér-Type Fractional Integral Inequalities for Weighted Fractional Integrals of a Function with Respect to Another Function
Fejér-type fractional integral inequalities are presented in this subsection concerning the weighted fractional integrals of a function with respect to another function.
Definition 18
([62]). Let and ψ be an increasing positive monotonic function on the interval with a continuous derivative on the interval with Then, the left-side and right-side of the weighted fractional integrals of a function Q with respect to another function on are defined by
and
where and
Theorem 64
([63]). Let be an integrable convex function with and be an integrable, positive, and weighted symmetric function with respect to If ψ an increasing and positive function on and is continuous on then, we have for :
where
Theorem 65
([63]). Let be an integrable convex function with for and be an integrable, positive, and weighted symmetric function with respect to If is harmonically convex on ψ an increasing and positive function on and ψ is continuous on then we have for :
where and
Theorem 66
([63]). Let be an integrable convex function with for and be an integrable, positive and weighted symmetric function with respect to If is harmonically convex on for ψ an increasing and positive function on and ψ is continuous on then, we have for :
where are defined in Theorem 65 and
Theorem 67
([64]). Assume that is a convex function with and be an integrable, positive and weighted symmetric function with respect to If ψ is an increasing and positive function on and is continuous on then, we have for :
Theorem 68
([65]). Let be an integrable harmonically convex function with and is an integrable, positive and weighted symmetric function with respect to If ψ is an increasing and positive function on and is continuous on then, we have for :
where
Theorem 69
([65]). Let with be continuous with a derivative such that and is harmonically convex on and let be an integrable, positive non-negative and weighted symmetric function with respect to If ψ an increasing and positive function on and is continuous on then, we have for :
where
with
Theorem 70
([65]). Assume that all the conditions of Theorem 69 hold and is harmonically convex on with Then, we have:
where , and are defined in Theorem 69
Theorem 71
([65]). Assume that all the conditions of Theorem 69 hold and are harmonically convex on with Then, we have:
where
6. Fejér-Type Fractional Integral Inequalities for Exponential Kernel
In the following, we give the definition of a fractional integral with an exponential kernel and present Fejér-type fractional integral inequalities for this new fractional integral.
Definition 19
([66]). Let The left and right fractional integrals of order are, respectively, defined by
and
Theorem 72
([66]). Let be a convex and integrable function with . If is non-negative, integrable and symmetric with respect to , then the fractional inequality is given as:
Theorem 73
([67]). Let be convex function with and If is non-negative, integrable and symmetric to , then is convex and monotonically increasing on and the fractional inequality is given as:
with and defined by
Theorem 74
([67]). Let be convex function with and If is non-negative, integrable and symmetric to , then is convex and monotonically increasing on and the fractional inequality is given as:
with and defined by
Theorem 75
([67]). Let be a positive convex function with and If is non-negative, integrable and symmetric to , then is convex and monotonically increasing on and the fractional inequality is given as:
with and
Theorem 76
([67]). Let be a positive convex function with and If is non-negative, integrable and symmetric to , then is convex and monotonically increasing on and the fractional inequality is given as:
with and defined by
Now, we prove both the first and second kind Fejér-type inequalities in a different approach.
Theorem 77
([68]). Let with be a convex function. If is a convex symmetric and integrable function with respect to then for then the fractional inequality is given as:
Theorem 78
([68]). Let with be a convex function. If is a convex symmetric and integrable function with respect to then for then the fractional inequality is given as:
Theorem 79
([68]). Let with be a convex function. Then for the fractional inequality is given as:
In the next we present Hadamard-Fejér type inequalities of both first and second kind for harmonically convex functions. Let us begin with the Hadamard-Fejér type inequality of the first kind.
Theorem 80
([68]). Let with be a harmonically convex function. If is a harmonically symmetric and integrable function with respect to then for the fractional inequality is given as:
where
Theorem 81
([68]). Let with be a harmonically convex function. Then, for we have:
Theorem 82
([68]). Let with be a harmonically convex function. If is a harmonically symmetric and integrable function with respect to then for the fractional inequality is given as:
The Fejeér–Hadamard–Mercer-type inequality for harmonically convex function is presented in the next result.
Theorem 83
([69]). Let be a harmonically convex function for with If and is non-negative, integrable, and harmonically symmetric with respect to then:
for all and
7. Fejér-Type Fractional Integral Inequalities via Hadamard Fractional Integral
In this section, Fejér-type fractional integral inequalities concerning the Hadamard fractional integral are presented.
Definition 20
([25]). The left-sided and right-sided Hadamard fractional integrals of order of function Q are defined by
and
Definition 21
([70]). A function is said to be geometric–arithmetically convex (GA-convex) on if
for any and
Theorem 84
([71]). Assume that is a -convex function such that where with and If is non-negative, integrable and geometrically symmetric with respect to then the fractional integral inequalities are given as:
Theorem 85
([71]). Assume that Q and g are defined as in Theorem 84. If is -convex on , then fractional integral inequality is given as:
Theorem 86
([71]). Assume that Q and g are defined as in the Theorem 84. If is -convex on , then fractional integral inequality is given as:
if and and
if and
Definition 22
([72]). The function is said to be GA-s-convex (geometric-arithmetically convex) on if, for every and , we have
We will use the notations
Theorem 87
([73]). Let be a differentiable mapping on and is GA-convex on with If is a continuous positive mapping and geometrically symmetric with respect to (i.e., holds for all then
where
Theorem 88
([73]). Let be a differentiable mapping on and is GA-convex on with and If g is as in Theorem 87, then the following inequality holds:
where and
Theorem 89
([74]). Let be a differentiable function on the interior of I, such that where and If is GA-s-convex on is continuous and geometrically symmetric with respect to then the following inequality for fractional integrals holds:
where
Theorem 90
([74]). Let be a differentiable function on the interior of I, such that where and If is GA-s-convex on is continuous and geometrically symmetric with respect to then the fractional inequalities are given as:
where
Theorem 91
([74]). Let Q and g be as in Theorem 90. If is GA-s-convex on then the fractional inequality is given as:
with and and
An important generalization of Hadamard fractional integrals is the Hadamard k-fractional integral operators.
Definition 23
([75]). Let . The left-sided and right-sided Hadamard k-fractional integrals of order and of function Q are defined by
and
where is the k-Gamma function defined by
Theorem 92
([76]). Assume that is a differentiable function on such that where with and If is a continuous, positive function, geometrically symmetric with respect to and is -s-convex on then the fractional inequality is given as:
where
Theorem 93
([76]). Assume that is a differentiable function on such that where with and If is a continuous, positive function, geometrically symmetric with respect to and is -s-convex on then the following k-Hadamard fractional integral inequality with holds:
where
Fejér-Type Fractional Integral Inequalities for Quasi-Geometrically Convex Functions
Definition 24
([77]). A function is said to be quasi-geometrically convex on I if
for all and
We give some Fejér-type fractional integral inequalities for quasi-geometrically convex functions.
Theorem 94
([78]). Let be a differentiable function on the interior of I, such that where and If is quasi-geometrically convex on is continuous and geometrically symmetric with respect to then the fractional inequality is given as:
where
Theorem 95
([78]). Let be a differentiable function on the interior of I, such that where and If is quasi-geometrically convex on is continuous and geometrically symmetric with respect to then the fractional inequality is given as:
Theorem 96
([78]). Under the assumptions of Theorem 95, where is quasi-geometrically convex on the fractional inequality is given as:
and
with and
Theorem 97
([79]). Let be a differentiable function on the interior of I, such that where and If is quasi-geometrically convex on is continuous and geometrically symmetric with respect to then the fractional inequality is given as:
Theorem 98
([79]). Let be a differentiable function on the interior of I, such that where and If is quasi-geometrically convex on is continuous and geometrically symmetric with respect to then the fractional inequality is given as:
Theorem 99
([79]). Let Q and g be as in Theorem 98. If is quasi-geometrically convex on then the fractional inequality is given as:
with and
8. Fejér-Type Fractional Integral Inequalities via Raina Integral
In [80], Raina introduced a class of functions formally defined by
where the coefficients ) are a bounded sequence of real positive numbers. In [81], the following left-sided and right-sided fractional integral operators are defined, respectively, as:
Now, we present Fejér-type fractional integral inequalities via the Raina integral.
Theorem 100
([82]). Let be a differentiable mapping on and with and continuous. If is convex on then the fractional inequality is given as:
where and
Theorem 101
([82]). Let Q and g be as in Theorem 100. If is convex on then the fractional inequality is given as:
where is defined in Theorem 100 and
Theorem 102
([82]). Let Q and g be as in Theorem 100. If is convex on then the fractional inequality is given as:
where and
Theorem 103
([83]). Let be a differentiable mapping on and with and continuous and symmetric to . If is convex on then the fractional inequality is given as:
where and
Theorem 104
([83]). Let Q and g be as in Theorem 103. If is convex on then the fractional inequality is given as:
where and
Theorem 105
([83]). Under the assumptions of Theorem 103, the fractional inequality is given as:
where and
9. Fejér-Type Fractional Integral Inequalities via Conformable Integrals
Definition 25
([84]). The left and right fractional conformable integrals of order are defined by
Definition 26
([85]). We say that the function is symmetrized convex on the interval if the symmetrical transform is convex on
Definition 27
([86]). Let I be a non-empty interval on Then, a function is called Wright-quasi-convex on I if
for all and
Definition 28
([86]). Let I and J be intervals on with Also, let be a function and a function with . Then, Q is called h-convex on I if
for all and
Definition 29
([86]). Let h be the function in Definition 28. A function is called h-symmetrized convex on the interval if the symmetrical transform is h-convex on
We present certain Fejér-type fractional integral inequalities involving the above defined fractional integral operators.
Theorem 106
([86]). Let Also, let be an interval on a symmetrized convex and integrable function and be integrable and symmetric to Then,
Theorem 107
([86]). Assume that the conditions of Theorem 106 hold. Then
Theorem 108
([86]). Let Also, let be an interval on be an integrable function. Also, is Wright-quasi-convex and integrable on and is integrable and symmetric to Then,
Theorem 109
([86]). Assume that the conditions of Theorem 108 hold. Then
Theorem 110
([86]). Assume that the function is h-symmetrized convex on the interval with h being integrable on and Q is integrable on and be integrable and symmetric to Then, we have
In the sequence, we give some more Fejér-Type fractional integral inequalities via the fractional conformable integral operators for p-convex functions.
Definition 30
([87]). A function is said to be p-convex, if
for all and
Theorem 111
([88]). Assume that is a p-convex function, and with If and is non-negative, integrable and p-symmetric with respect to then, the fractional inequalities are given as:
with and
with
Definition 31
([89]). Let Then, the left- and right-sided conformable fractional integrals of order are given by
Now, we give Fejér-type fractional integral inequalities for convex functions via the conformable fractional integral defined above.
Theorem 112
([90]). Assume that is a convex function, with If and is non-negative, integrable and symmetric with respect to then the fractional inequalities are given as:
for
Theorem 113
([90]). Assume that is a differentiable function on , and with If is convex on and is continuous and symmetric with respect to then the fractional inequalities are given as:
for
Theorem 114
([90]). Assume that is a differentiable function on , and with If is convex on and is continuous and symmetric with respect to then the fractional inequalities are given as:
for and
Theorem 115
([90]). Assume that is a differentiable function on , and with If is convex on and is continuous and symmetric with respect to then the fractional inequalities are given as:
for and
Theorem 116
([91]). Assume that is a differentiable function on , and with If is convex on and is continuous and symmetric with respect to then the fractional inequalities are given as:
for
Theorem 117
([91]). Assume that is a differentiable function on , and with If is convex on for and is continuous and symmetric with respect to then the fractional inequalities are given as:
for and
Theorem 118
([91]). Assume that is a differentiable function on , and with If is convex on for and is continuous and symmetric with respect to then the fractional inequalities are given as:
for
Theorem 119
([92]). Assume that is a p-convex function, and If and is non-negative, integrable and p-symmetric with respect to then the fractional inequalities are given as:
with and
with
Definition 32
([93]). Let be a given function. Then, the subset D of a real linear space X will be called k-convex if for all and
Let be two given functions and suppose that be a k-convex set. Then, a function is -convex, if for all and
Theorem 120
([94]). Let be a -convex function with Assume that such that and let be a non-negative function which is symmetric with respect to Then, the fractional inequalities are given as:
for
10. Fejér-Type Fractional Integral Inequalities via Non-Conformable Fractional Integral
Definition 33
([95]). For each and , non-conformable fractional integral operator is then given b y
for every and .
Definition 34
([95]). For each function , then left and right non-conformable fractional integral operators are stated by
for every and .
Theorem 121
([96]). Suppose is an h–preinvex function, condition-C for Φ holds and , and , is symmetric with respect to and . Then,
Theorem 122
([96]). Let be an open invex subset with respect to and with . Suppose that is a differentiable mapping on and is differentiable and symmetric to . If is h-preinvex on and , then
Theorem 123
([96]). Let be an open invex subset with respect to and with Suppose that is a differentiable mapping on and is differentiable and symmetric to . If , , is h-preinvex on and , then one has:
Theorem 124
([96]). Let be an open invex subset with respect to and with Suppose that is a differentiable mapping on and is differentiable and symmetric to . If , is h-preinvex on and , then
where .
The next theorems include Fejér-type integral inequalities via a non-conformable fractional integral operator for generalized -preinvex functions.
Definition 35
([97]). Assume that , and . Then, Q is said to be generalized -preinvex function if
for all and
Theorem 125
([98]). Suppose is an -preinvex function, Condition C for Φ holds, , , and , is symmetric with respect to . Then
Theorem 126
([98]). Assume that and , such that is an open m-invex subset with respect to Φ and with . Suppose that is a differentiable mapping on and is differentiable and symmetric to . If is generalized -preinvex on , then
Theorem 127
([98]). Assume that and are defined as in Theorem 122. If —where —is generalized -preinvex on , then one has
Theorem 128
([98]). Assume that and are defined as in Theorem 122. If —where —is generalized -preinvex on , then
where .
11. Fejér-Type Fractional Integral Inequalities via Katugampola Fractional Integral
Definition 36
([99]). Let be a finite interval. Then, the left- and right-side Katugampola fractional integrals of order of are defined by
with and if the integrals exist. Here, denote the space of those complex-valued Lebesque measurable functions Q on for which where for and if
Fejér-type fractional integral inequalities via Katugampola fractional integral are given in the next theorems.
Theorem 129
([100]). Let be convex function with and . Then, is also convex and . If is non-negative and integrable function, then
with and
Theorem 130
([100]). Let be differentiable function on and with . Then is also differentiable and . If is convex and is a continuous function, then
with and where and .
In the following, we present a Fejér-type fractional integral inequality for -convex functions via a Katugampola fractional integral.
Theorem 131
([101]). Let be a -convex function with Assume that such that and If is a non-negative function which is symmetric with respect to and is non-negative and integrable, then
for and
12. Fejér-Type Fractional Integral Inequalities for Invex Functions
Fejér-Type Fractional Integral Inequalities for -Convex Functions
Definition 37
([102]). A set is invex with respect to a real bifunction if
Definition 38
([102]). Let be an invex set with respect to Consider and The function Q is said to be -convex if
for all and
Fejér-type fractional integral inequalities for -convex functions are presented in the following theorems.
Theorem 132
([102]). Let be an invex set with respect to such that
for all and Also, let be an -convex function, where is an integrable bifunction on For any with suppose that and the function are integrable and symmetric to Then,
Theorem 133
([102]). Let be an invex set with respect to let be an -convex function, where is an integrable bifunction on For any with suppose that is integrable and symmetric to and Then
Definition 39
([103]). Let be an invex set with respect to A function is said to be -preinvex with respect to Φ for every and if
Theorem 134
([103]). Let be an open invex subset with respect to and with where Suppose is a differentiable mapping on such that If is a continuous mapping and symmetric to and is -preinvex on then
Theorem 135
([103]). Assume that Q is as in Theorem 134 and is -preinvex on then fractional integral inequality is given as:
where
13. Fejér-Type Fractional Integral Inequalities via -Calculus
We first give the basic concepts of -calculus which will be used in our work. Throughout this section, we let , , and be constants.
Definition 40
([104,105]). Let be a continuous function. The -derivative of the function Q on at x is defined by
Since Q is continuous function, we have .
We say that Q is -differentiable on I provided that exists for all . In Definition 40, if , then , where is defined by:
Furthermore, if , then which is the q-derivative of the function Q.
Definition 41
([104,105]). If is a continuous function, then the -integral of the function Q for is defined by
Moreover, if , then the -integral is defined by
We say that Q is -integrable on I provided that exists for all . Note that, if and , then Definition 41 reduces to the q-integral of function Q.
Next, we give some Fejér-type fractional integral inequalities by using the -integral.
Theorem 136
([106]). If is a twice -differentiable function such that there exist real constants m and M so that , then
and
Theorem 137
([106]). Let be a twice -differentiable function such that there exist real constants m and M so that . If is non-negative, -integrable on I, and symmetric about , then
and
Theorem 138
([106]). If is a twice -differentiable function with -integrable on I and , then
and
Theorem 139
([106]). If is a twice -differentiable function with -integrable on I and , then
Theorem 140
([106]). If is a twice -differentiable function with -integrable on I and , then
14. Conclusions
Fractional calculus has significant importance and several uses in applied mathematics. Researchers and mathematicians in the theory of inequalities have employed fractional calculus operators to investigate and explore specific estimations and developments. Fractional integral operators are remarkable and meaningful applicable tools for generalizing classical integral inequalities. Due to its many applications and uses, fractional calculus has captured the attention and excitement of many mathematicians. Considering its utility in the mathematical modeling of various complicated and non-local nonlinear systems, fractional calculus has developed to be a crucial field of investigation. It is critical while exploring optimization problems since it has a variety of advantageous inequalities.
The main intention and aim of this review paper were to deliver an in-depth and current assessment of Fejér-type integral inequalities. We offered numerous results of fractional Fejér-type integral inequality via convexity.
We anticipate that this study will inspire as well as offer a forum for scholars pursuing Fejér-type inequalities to acquire information about prior research on the topic before drawing fresh conclusions. This review paper’s future research is promising and might motivate numerous additional studies.
Author Contributions
Conceptualization, M.T. and S.K.N.; methodology, M.T., S.K.N. and A.A.S.; validation, M.T., S.K.N. and A.A.S.; supervision, S.K.N. and A.A.S.; formal analysis, M.T., S.K.N. and A.A.S.; writing—original draft preparation, M.T. and S.K.N. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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