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Review

A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators

1
Department of Basic Sciences and Related Studies, Mehran UET, Jamshoro 76062, Pakistan
2
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
*
Author to whom correspondence should be addressed.
Axioms 2023, 12(7), 719; https://doi.org/10.3390/axioms12070719
Submission received: 20 June 2023 / Revised: 15 July 2023 / Accepted: 21 July 2023 / Published: 24 July 2023
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

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, ( θ , h m ) p -convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, ( k p ) -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 ( p , q ) -calculus are also included.

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:
Q w 1 + w 2 2 1 w 2 w 1 w 1 w 2 Q ( x ) d x Q ( w 1 ) + Q ( w 2 ) 2 .
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 Q : [ w 1 , w 2 ] R is a convex function. Then, the inequality
Q w 1 + w 2 2 w 1 w 2 g ( x ) d x w 1 w 2 Q ( x ) g ( x ) d x Q ( w 1 ) + Q ( w 2 ) 2 w 1 w 2 g ( x ) d x
holds, where g : [ a , b ] R is non-negative, integrable and symmetric to w 1 + w 2 2 .
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, ( θ , h m ) p -convex functions, and h-preinvex function. In Section 3, we present Fejér-type fractional integral inequalities using the ( k p ) -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 ( p , q ) -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 Q L [ w 1 , w 2 ] . Then, Riemann–Liouville integrals in the left and right sense J w 1 + α Q and J w 2 α Q , α > 0 , w 1 0 are stated by
J w 1 + α Q ( x ) = 1 Γ ( α ) w 1 x ( x t ) α 1 Q ( t ) d t , x > w 1 ,
and
J w 2 α Q ( x ) = 1 Γ ( α ) x w 2 ( t x ) α 1 Q ( t ) d t , x < w 2 ,
respectively. Here, Γ ( α ) represents the Euler Gamma function and J w 1 + 0 Q ( x ) = J w 2 0 Q ( x ) = Q ( x ) .

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 Q : I R be a differentiable function on the interior I of I , such that Q L [ w 1 , w 2 ] for w 1 , w 2 I with w 1 < w 2 and g : [ w 1 , w 2 ] R be a continuous function. If | Q | is convex on [ w 1 , w 2 ] , then, for any α > 0 and x [ w 1 , w 2 ] , the fractional inequality is given as:
Γ ( α ) Q ( x ) J x + α g ( w 2 ) + J x α g ( w 1 ) J x + α ( Q g ) ( w 2 ) + J x α ( Q g ) ( w 1 ) ( x w 1 ) α + 1 g [ w 1 , x ] Γ ( α + 3 ) ( α + 1 ) | Q ( x ) | + Q ( w 1 ) | + ( w 2 x ) α + 1 g [ x , w 2 ] Γ ( α + 3 ) ( α + 1 ) | Q ( x ) | + Q ( w 2 ) | .
Theorem 3 
([26]). Let Q : I R be a differentiable function on the interior I such that Q L [ w 1 , w 2 ] for w 1 , w 2 I with w 1 < w 2 and g : [ w 1 , w 2 ] R be a continuous function. If | Q | q is convex on [ w 1 , w 2 ] for some fixed q > 1 with 1 p + 1 q = 1 , then, for any α > 0 and x [ w 1 , w 2 ] the fractional inequality is given as:
Γ ( α ) Q ( x ) J x + α g ( w 2 ) + J x α g ( w 1 ) J x + α ( Q g ) ( w 2 ) + J x α ( Q g ) ( w 1 ) g [ w 1 , x ] Γ ( α + 1 ) ( x w 1 ) α + 1 ( 1 + α p ) 1 p | Q ( x ) | q + Q ( w 1 ) | q 2 1 q + g [ x , w 2 ] Γ ( α + 1 ) ( w 2 x ) α + 1 ( 1 + α p ) 1 p | Q ( x ) | q + Q ( w 2 ) | q 2 1 q .
Theorem 4 
([26]). Let Q : I R be a differentiable function on the interior I such that Q L [ w 1 , w 2 ] for w 1 , w 2 I with w 1 < w 2 and g : [ w 1 , w 2 ] R be a continuous function. If | Q | q is convex on [ w 1 , w 2 ] for some fixed q 1 , then, for any α > 0 and x [ w 1 , w 2 ] , the fractional inequality is given as:
Γ ( α ) Q ( x ) J x + α g ( w 2 ) + J x α g ( w 1 ) J x + α ( Q g ) ( w 2 ) + J x α ( Q g ) ( w 1 ) g [ w 1 , x ] Γ ( α + 1 ) ( x w 1 ) α + 1 ( α + 1 ) | Q ( x ) | q + Q ( w 1 ) | q 2 1 q + g [ x , w 2 ] Γ ( α + 1 ) ( w 2 x ) α + 1 ( α + 1 ) | Q ( x ) | q + Q ( w 2 ) | q 2 1 q .
Theorem 5 
([27]). Let Q : [ w 1 , w 2 ] R be a differentiable mapping on ( w 1 , w 2 ) and Q [ w 1 , w 2 ] with w 1 < w 2 . If | Q | is convex on [ w 1 , w 2 ] and g : [ w 1 , w 2 ] R is continuous and symmetric to w 1 + w 2 2 , the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 1 α ( Q g ) ( w 1 ) | g Γ ( α ) ( w 2 w 1 ) α + 1 1 2 α ( α + 1 ) + 1 α + 1 1 2 α | Q ( w 1 ) | + | Q ( w 2 ) | , α > 0 ,
where g = sup { | g ( t ) | ; t [ w 1 , w 2 ] } .
Theorem 6 
([27]). Suppose that all the conditions of Theorem 5 hold. Then, the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α g ( Q g ) ( w 2 ) + J w 1 α g ( Q g ) ( w 1 ) | g Γ ( α ) ( w 2 w 1 ) α + 1 | Q ( w 1 ) | + | Q ( w 2 ) | 8 , α > 0 .
Theorem 7 
([27]). Suppose that all the conditions of Theorem 5 hold. Additionally, we assume that | Q | q , q > 1 is convex on [ w 1 , w 2 ] . Then, the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + α g ( w 2 ) + J w 1 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | g ( w 2 w 1 ) α + 1 4 1 p + 1 1 p 3 | Q ( w 1 ) | q + | Q ( w 2 ) | q 4 1 q + | Q ( w 1 ) | q + 3 | Q ( w 2 ) | q 4 1 q ,
where 1 p + 1 q = 1 .
Theorem 8 
([28]). Let Q : I R R be a differentiable mapping on I , w 1 , w 2 I with w 1 < w 2 , and let g : [ w 1 , w 2 ] R be continuous on [ w 1 , w 2 ] . If | Q | is convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| w 1 w 2 g ( u ) d u α [ Q ( w 1 ) + Q ( w 2 ) ] α w 1 w 2 w 1 t g ( u ) d u α 1 g ( t ) Q ( t ) d t α w 1 w 2 t w 2 g ( u ) d u α 1 g ( t ) Q ( t ) d t | g ( w 2 w 1 ) α + 1 α + 1 [ | Q ( w 1 ) | + | Q ( w 2 ) | ] .
Theorem 9 
([28]). Let Q : I R R be a differentiable mapping on I , w 1 , w 2 I with w 1 < w 2 and let g : [ w 1 , w 2 ] R be continuous on [ w 1 , w 2 ] . If | Q | q is convex on [ w 1 , w 2 ] , q > 1 , then the fractional inequality is given as:
| w 1 w 2 g ( u ) d u α [ Q ( w 1 ) + Q ( w 2 ) ] α w 1 w 2 w 1 t g ( u ) d u α 1 g ( t ) Q ( t ) d t α w 1 w 2 t w 2 g ( u ) d u α 1 g ( t ) Q ( t ) d t | 2 g α ( w 2 w 1 ) α + 1 ( α p + 1 ) 1 p | Q ( w 1 ) | q + | Q ( w 2 ) | q 2 1 q ,
where α > 0 and 1 p + 1 q = 1 .
Theorem 10 
([29]). Let Q : [ w 1 , w 2 ] R be a convex function with w 1 < w 2 and Q L [ w 1 , w 2 ] . If g : [ w 1 , w 2 ] R is non-negative, integrable, and symmetric to w 1 + w 2 2 , then inequality in the frame of fractional operator is given as:
Q w 1 + w 2 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) Q ( w 1 ) + Q ( w 2 ) 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) , α > 0 .
Theorem 11 
([29]). Let Q : I R R be a differentiable mapping on I and Q L [ w 1 , w 2 ] with w 1 < w 2 . If | Q | is a convex function on [ w 1 , w 2 ] and g : [ w 1 , w 2 ] R is continuous and symmetric to w 1 + w 2 2 , then inequality in the frame of the fractional operator is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g ( α + 1 ) Γ ( α + 1 ) 1 1 2 α | Q ( w 1 ) | + | Q ( w 2 ) | , α > 0 .
Theorem 12 
([30]). Assume that w 1 , w 2 R with w 1 < w 2 and Q : I R is a differentiable function on I such that Q L [ w 1 , w 2 ] and g : [ w 1 , w 2 ] R is continuous. If | Q | is a convex function on I , then an inequality in the frame of the fractional operator is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 α g ( w 1 ) + J w 1 + w 2 2 + α g ( w 2 ) J w 1 + w 2 2 α ( Q g ) ( w 1 ) + J w 1 + w 2 2 + α ( Q g ) ( w 2 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 ( α + 1 ) Γ ( α + 1 ) [ | Q ( w 1 ) | + | Q ( w 2 ) | ] ,
where g = sup { | g ( t ) | , t [ w 1 , w 2 ] } .
Theorem 13 
([30]). Assume that Q is as in Theorem 12 and g : [ w 1 , w 2 ] R is continuous. If | Q | q , q > 1 is a convex function on I , then an inequality in the frame of the fractional operator is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 α g ( w 1 ) + J w 1 + w 2 2 + α g ( w 2 ) J w 1 + w 2 2 α ( Q g ) ( w 1 ) + J w 1 + w 2 2 + α ( Q g ) ( w 2 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 + 2 q ( α p + 1 ) 1 / p Γ ( α + 1 ) × 3 | Q ( w 1 ) | q + | Q ( w 2 ) | q 1 q + | Q ( w 1 ) | q + 3 | Q ( w 2 ) | q 1 q ,
where 1 p + 1 q = 1 .

2.2. Fejér-Type Fractional Integral Inequalities for s-Convex Functions

Definition 2 
([31]). A function Q : [ 0 , ) R is said to be s-convex in the second sense if
Q ( λ x + ( 1 λ ) y ) λ s Q ( x ) + ( 1 λ ) s Q ( y )
for all x , y [ 0 , ) , λ [ 0 , 1 ] and for some fixed s ( 0 , 1 ] .
In the next theorems, Fejér-type fractional integral inequalities for s-convex functions are presented.
Theorem 14 
([32]). Let Q : I R R be a differentiable mapping on I , w 1 , w 2 I with w 1 < w 2 and let g : [ w 1 , w 2 ] R be continuous on [ w 1 , w 2 ] . If | Q | is s-convex in the second sense on [ w 1 , w 2 ] , then the fractional inequality is given as:
| w 1 w 2 g ( u ) d u α [ Q ( w 1 ) + Q ( w 2 ) ] α w 1 w 2 w 1 t g ( u ) d u α 1 g ( t ) Q ( t ) d t α w 1 w 2 t w 2 g ( u ) d u α 1 g ( t ) Q ( t ) d t | g ( w 2 w 1 ) α 1 A ( α , s ) [ | Q ( w 1 ) | + | Q ( w 2 ) | ]
where
A ( α , s ) = 1 α + s + 1 1 1 2 α + s + B 1 / 2 ( s + 1 , α + 1 ) B 1 / 2 ( α + 1 , s + 1 ) ,
and B x is the incomplete beta function defined as follows
B x ( m , n ) = 0 x t m 1 ( 1 t ) n 1 d t , m , n > 0 , 0 < x < 1 .
Theorem 15 
([32]). Let Q : I R R be a differentiable mapping on I , w 1 , w 2 I with w 1 < w 2 and let g : [ w 1 , w 2 ] R be continuous on [ w 1 , w 2 ] . If | Q | q is s-convex in the second sense on [ w 1 , w 2 ] , q > 1 , then the fractional inequality be given as:
| w 1 w 2 g ( u ) d u α [ Q ( w 1 ) + Q ( w 2 ) ] α w 1 w 2 w 1 t g ( u ) d u α 1 g ( t ) Q ( t ) d t α w 1 w 2 t w 2 g ( u ) d u α 1 g ( t ) Q ( t ) d t | g α ( w 2 w 1 ) α + 1 ( α p + 1 ) 1 p 1 1 2 α p 1 p | Q ( w 1 ) | q + | Q ( w 2 ) | q s + 1 1 q ,
where α > 0 and 1 p + 1 q = 1 .
Theorem 16 
([33]). Let Q : I [ 0 , ) R be a differentiable function on I and Q L [ w 1 , w 2 ] where w 1 < w 2 and g : [ w 1 , w 2 ] R is continuous and symmetric to w 1 + w 2 2 . If | Q | is s-convex in the second sense on [ w 1 , w 2 ] for some fixed s ( 0 , 1 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g ( α + s + 1 ) Γ ( α + 1 ) ( 1 ( α + s + 1 ) B 1 / 2 ( s + 1 , α + 1 ) B 1 / 2 ( α + 1 , s + 1 ) × | Q ( w 1 ) | + | Q ( w 2 ) | .
Theorem 17 
([33]). Let Q : I [ 0 , ) R be differentiable function on I and Q L [ w 1 , w 2 ] where w 1 < w 2 and g : [ w 1 , w 2 ] R be continuous and symmetric to w 1 + w 2 2 . If | Q | q , q > 1 is s-convex in the second sense on [ w 1 , w 2 ] for some fixed s ( 0 , 1 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | 2 ( w 2 w 1 ) α + 1 g ( α + 1 ) 1 1 q ( α + s + 1 ) 1 q Γ ( α + 1 ) 1 1 2 α 1 1 q | Q ( w 1 ) | q + | Q ( w 2 ) | q 2 1 q × 1 ( α + s + 1 ) B 1 / 2 ( s + 1 , α + 1 ) B 1 / 2 ( α + 1 , s + 1 ) 1 q .
Theorem 18 
([33]). Assume that the conditions of Theorem 17 hold true. Then:
(i) 
For α > 0 , we have:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | 2 1 p ( w 2 w 1 ) α + 1 g ( α p + 1 ) 1 p Γ ( α + 1 ) 1 1 2 α 1 p | Q ( w 1 ) | q + | Q ( w 2 ) | q s + 1 1 q .
(ii) 
For 0 < α < 1 and 1 p + 1 q = 1 , we have:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 + α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g ( α p + 1 ) 1 p Γ ( α + 1 ) | Q ( w 1 ) | q + | Q ( w 2 ) | q s + 1 1 q .
Theorem 19 
([34]). Let Q : I [ 0 , ) R be a differentiable function on I and Q L [ w 1 , w 2 ] , where w 1 < w 2 and g : [ w 1 , w 2 ] R are continuous. If | Q | is s-convex in the second sense on [ w 1 , w 2 ] for some fixed s ( 0 , 1 ] , then the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 + α Q ( w 2 ) + J w 1 + w 2 2 α Q ( w 1 ) J w 1 + w 2 2 + α ( Q g ) ( w 2 ) + J w 1 + w 2 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g Γ ( α + 1 ) B 1 / 2 ( α + 1 , s + 1 ) + 1 2 α + s + 1 ( α + s + 1 ) | Q ( w 1 ) | + | Q ( w 2 ) | .
Theorem 20 
([34]). Let Q : I [ 0 , ) R be differentiable function on I and Q L [ w 1 , w 2 ] where w 1 < w 2 and g : [ w 1 , w 2 ] R be continuous. If | Q | q , q 1 is s-convex in the second sense on [ w 1 , w 2 ] for some fixed s ( 0 , 1 ] , then the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 + α Q ( w 2 ) + J w 1 + w 2 2 α Q ( w 1 ) J w 1 + w 2 2 + α ( Q g ) ( w 2 ) + J w 1 + w 2 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 + 1 q ( α + 1 ) ( α + s + 1 ) 1 q Γ ( α + 1 ) × [ ( 2 α + 2 ( α + 1 ) ( α + s + 1 ) B 1 / 2 ( α + 1 , s + 1 ) | Q ( w 1 ) | q + 2 1 s ( α + 1 ) | Q ( w 2 ) | q ) 1 q + ( 2 1 s ( α + 1 ) | Q ( w 1 ) | q + 2 α + 2 ( α + 1 ) ( α + s + 1 ) B 1 / 2 ( α + 1 , s + 1 ) | Q ( w 2 ) | q ) 1 q ] .
Theorem 21 
([34]). Let Q : I [ 0 , ) R be differentiable function on I and Q L [ w 1 , w 2 ] where w 1 < w 2 and g : [ w 1 , w 2 ] R be continuous. If | Q | q , q > 1 is s-convex in the second sense on [ w 1 , w 2 ] for some fixed s ( 0 , 1 ] , then the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 + α Q ( w 2 ) + J w 1 + w 2 2 α Q ( w 1 ) J w 1 + w 2 2 + α ( Q g ) ( w 2 ) + J w 1 + w 2 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 + 1 q ( α p + 1 ) 1 p ( s + 1 ) 1 q Γ ( α + 1 ) × ( ( 2 s + 1 1 ) | Q ( w 1 ) | q + | Q ( w 2 ) | q ) 1 q + ( | Q ( w 1 ) | q + ( 2 s + 1 1 ) | Q ( w 2 ) | q ) 1 q .
Theorem 22 
([35]). Let Q : I [ 0 , ) R be a differentiable function on I such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I with w 1 < w 2 . If | Q | is s-convex in the second sense on [ w 1 , w 2 ] , then inequality in the frame of the fractional operator is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 α g ( w 1 ) + J w 1 + w 2 2 + α g ( w 2 ) J w 1 + w 2 2 α ( Q g ) ( w 1 ) + J w 1 + w 2 2 + α ( Q g ) ( w 2 ) | ( w 2 w 1 ) α + 1 g Γ ( α + 1 ) B 1 / 2 ( α + 1 , s + 1 ) + 1 2 α + s + 1 ( α + s + 1 ) [ | Q ( w 1 ) | + | Q ( w 2 ) | ] .
Theorem 23 
([35]). Let Q : I [ 0 , ) R be a differentiable function on I such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I with w 1 < w 2 and g : [ w 1 , w 2 ] R is continuous. If | Q | q is s-convex in the second sense on [ w 1 , w 2 ] , then an inequality in the frame of fractional operator is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 α g ( w 1 ) + J w 1 + w 2 2 + α g ( w 2 ) J w 1 + w 2 2 α ( Q g ) ( w 1 ) + J w 1 + w 2 2 + α ( Q g ) ( w 2 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 + 1 q ( α + 1 ) ( α + 2 ) 1 q ( α + s + q ) 1 q Γ ( α + 1 ) × { ( α + s + 1 ) ( α + 3 ) | Q ( w 1 ) | q + 2 1 s ( α + 1 ) ( α + 2 ) | Q ( w 2 ) | q 1 q + 2 1 s ( α + 1 ) ( α + 2 ) | Q ( w 1 ) | q + ( α + s + 1 ) ( α + 3 ) | Q ( w 2 ) | q 1 q } .

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 I R { 0 } is a real interval. A function Q : I R is harmonically s-convex if
Q x y t x + ( 1 t ) y t s Q ( y ) + ( 1 t ) s Q ( x )
for all x , y I , t [ 0 , 1 ] and s ( 0 , 1 ] .
Theorem 24 
([37]). Let Q : I ( 0 , ) R be a differentiable function on I , the interior of I , such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | is harmonically s-convex on [ w 1 , w 2 ] , g : [ w 1 , w 2 ] R is continuous and harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 / w 2 + α ( g h ) ( 1 / w 1 ) + J 1 / w 1 α ( g h ) ( 1 / w 2 ) J 1 / w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 / w 1 α ( Q g h ) ( 1 / w 2 ) | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α [ C 1 ( α ) | Q ( w 1 ) | + C 2 ( α ) | Q ( w 2 ) | ,
where
C 1 ( α ) = w 2 2 α + s + 1 2 F 1 2 , 1 ; α + s + 2 ; 1 w 1 w 2 β ( α + 1 , s + 1 ) w 2 2 2 F 1 ( 2 , α + 1 ; α + s + 2 ; 1 w 1 w 2 ) + β ( α + 1 , s + 1 ) 2 s 2 ( w 1 + w 2 ) 2 2 F 1 ( 2 , α + 1 ; α + s + 2 ; w 2 w 1 w 2 + w 1 ) C 2 ( α ) = w 2 2 α + s + 1 2 F 1 2 , α + s + 1 ; α + s + 2 ; 1 w 1 w 2 β ( s + 1 , α + 1 ) w 2 2 2 F 1 ( 2 , s + 1 ; α + s + 2 ; 1 w 1 w 2 ) + 1 2 s β ( s + 1 , α + 1 ) w 2 2 2 F 1 ( 2 , s + 1 ; α + s + 2 ; 1 2 1 w 1 w 2 ) ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .
Theorem 25 
([37]). Let Q : I ( 0 , ) R be a differentiable function on I , the interior of I , such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | q , q 1 is harmonically s-convex on [ w 1 , w 2 ] , g : [ w 1 , w 2 ] R is continuous and harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 / w 2 + α ( g h ) ( 1 / w 1 ) + J 1 / w 1 α ( g h ) ( 1 / w 2 ) J 1 / w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 / w 1 α ( Q g h ) ( 1 / w 2 ) | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α { C 3 1 1 q ( α ) C 4 ( α ) | Q ( w 1 ) | q + C 5 ( α ) | Q ( w 2 ) | q 1 q + C 6 1 1 q ( α ) C 7 ( α ) | Q ( w 1 ) | q + C 8 ( α ) | Q ( w 2 ) | q 1 q }
where
C 3 ( α ) = 2 ( w 1 + w 2 ) 2 α + 1 2 F 1 2 , α + 1 ; α + 3 ; w 2 w 1 w 1 + w 2 , C 4 ( α ) = ( w 1 + w 2 ) 2 2 s 1 ( α + 1 ) ( α + 2 ) 2 F 1 2 , α + 1 ; α + 3 ; w 2 w 1 w 1 + w 2 , C 5 ( α ) = w 2 2 α + s + 1 2 F 1 2 , 1 ; α + s + 1 ; α + s + 2 ; 1 w 1 w 2 β ( α + 1 , s + 1 ) w 2 2 2 F 1 ( 2 , s + 1 ; α + s + 2 ; 1 w 1 w 2 ) + 1 2 s + 1 β ( α + 1 , s + 1 ) w 2 2 2 F 1 ( 2 , s + 1 ; α + s + 2 ; 1 2 1 w 1 w 2 ) , C 6 ( α ) = w 2 2 α + 1 2 F 1 2 , 1 ; α + 2 ; 1 w 1 w 2 w 2 2 α + 1 2 F 1 2 , α + 1 ; α + 2 ; 1 w 1 w 2 + C 3 ( α ) , C 7 ( α ) = w 2 2 α + s + 1 2 F 1 2 , 1 ; α + s + 2 ; 1 w 1 w 2 β ( α + 1 , s + 1 ) w 2 2 2 F 1 ( 2 , α + 1 ; α + s + 2 ; 1 w 1 w 2 ) + C 4 ( α ) , C 8 ( α ) = β ( α + 1 , s + 1 ) w 2 2 2 F 1 ( 2 , s + 1 ; α + s + 2 ; 1 w 1 w 2 ) w 2 2 α + s + 1 2 F 1 2 , α + s + 1 ; α + s + 2 ; 1 w 1 w 2 + C 5 ( α ) ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .
Theorem 26 
([37]). Let Q : I ( 0 , ) R be a differentiable function on I , the interior of I , such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | q , q > 1 is harmonically s-convex on [ w 1 , w 2 ] , g : [ w 1 , w 2 ] R is continuous and harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 / w 2 + α ( g h ) ( 1 / w 1 ) + J 1 / w 1 α ( g h ) ( 1 / w 2 ) J 1 / w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 / w 1 α ( Q g h ) ( 1 / w 2 ) | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α { C 9 1 p ( α ) | Q ( w 1 ) | q + ( 2 s + 1 1 ) | Q ( w 2 ) | q 2 s + 1 ( s + 1 ) 1 q + C 10 1 p ( α ) ( 2 s + 1 1 ) | Q ( w 1 ) | q + | Q ( w 2 ) | q 2 s + 1 ( s + 1 ) 1 q }
where
C 9 ( α ) = w 1 + w 2 2 2 p 1 2 ( α p + 1 ) 2 F 1 2 p , α p + 1 ; α p + 2 ; w 2 w 1 w 1 + w 2 , C 10 ( α ) = w 2 2 p 1 2 ( α p + 1 ) 2 F 1 2 p , α p + 1 ; α p + 2 ; 1 2 1 w 1 w 2
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .

2.4. Fejér-Type Fractional Integral Inequalities for Quasi-Convex Functions

Definition 4 
([38]). A function Q : [ w 1 , w 2 ] R is said to be quasi-convex on [ w 1 , w 2 ] if
Q ( λ x + ( 1 λ ) y ) sup { Q ( x ) , Q ( y ) } ,
for all x , y [ w 1 , w 2 ] and λ [ 0 , 1 ] .
In the following, we present theorems including Fejér-type fractional integral inequalities for quasi-convex functions.
Theorem 27 
([39]). Let Q : I R be a differentiable mapping on I and Q L [ w 1 , w 2 ] with w 1 < w 2 and g : [ w 1 , w 2 ] R be continuous. If | Q | q is quasi convex on [ w 1 , w 2 ] , q > 1 , then the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 α g ( w 1 ) + J w 1 + w 2 2 + α g ( w 2 ) J w 1 + w 2 2 α ( Q g ) ( w 1 ) + J w 1 + w 2 2 + α ( Q g ) ( w 2 ) | ( w 2 w 1 ) α + 1 g 2 α ( α + 1 ) Γ ( α + 1 ) sup { | Q ( w 1 ) | q , | Q ( w 2 ) | q } 1 q ,
where α > 0 .
Theorem 28 
([39]). Suppose that all the conditions of Theorem 27 hold. Then, the fractional inequality is given as:
| Q w 1 + w 2 2 J w 1 + w 2 2 α g ( w 1 ) + J w 1 + w 2 2 + α g ( w 2 ) J w 1 + w 2 2 α ( Q g ) ( w 1 ) + J w 1 + w 2 2 + α ( Q g ) ( w 2 ) | ( w 2 w 1 ) α + 1 g 2 α ( α p + 1 ) 1 p Γ ( α + 1 ) sup { | Q ( w 1 ) | q , | Q ( w 2 ) | q } 1 q ,
where α > 0 , and 1 p + 1 q = 1 .
Theorem 29 
([39]). Let Q : I R be a differentiable mapping on I and Q L [ w 1 , w 2 ] with w 1 < w 2 and g : [ w 1 , w 2 ] R is continuous and symmetric to w 1 + w 2 2 . If | Q | is quasi convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | 2 ( w 2 w 1 ) α + 1 g ( α + 1 ) Γ ( α + 1 ) 1 1 2 α sup { | Q ( w 1 ) | , | Q ( w 2 ) | } ,
where α > 0 .
Theorem 30 
([39]). Let Q : I R be a differentiable mapping on I and Q L [ w 1 , w 2 ] with w 1 < w 2 and g : [ w 1 , w 2 ] R be continuous and symmetric to w 1 + w 2 2 . If | Q | q is quasi convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 α g ( w 2 ) + J w 2 α g ( w 2 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | 2 ( w 2 w 1 ) α + 1 g ( α + 1 ) Γ ( α + 1 ) 1 1 2 α sup { | Q ( w 1 ) | q , | Q ( w 2 ) | q } 1 q ,
where α > 0 .
Theorem 31 
([39]). Suppose that all the conditions of Theorem 30 hold. Then, the fractional inequality is given as:
(i) 
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | 2 1 p ( w 2 w 1 ) α + 1 g ( α p + 1 ) 1 p Γ ( α + 1 ) 1 1 2 α p 1 p sup { | Q ( w 1 ) | q , | Q ( w 2 ) | q } 1 q ,
where α > 0 .
(ii) 
| Q ( w 1 ) + Q ( w 2 ) 2 J w 1 α g ( w 2 ) + J w 2 α g ( w 1 ) J w 1 + α ( Q g ) ( w 2 ) + J w 2 α ( Q g ) ( w 1 ) | ( w 2 w 1 ) α + 1 g ( α p + 1 ) 1 p Γ ( α + 1 ) sup { | Q ( w 1 ) | q , | Q ( w 2 ) | q } 1 q ,
where α > 0 , and 1 p + 1 q = 1 .

2.5. Fejér-Type Fractional Integral Inequalities for Harmonically Convex Functions

Definition 5 
([40]). Assume that I R { 0 } is a real interval. A function Q : I R is harmonically convex, if
Q x y t x + ( 1 t ) y t Q ( y ) + ( 1 t ) Q ( x )
for all x , y I and t [ 0 , 1 ] .
Definition 6 
([40]). A function g : [ w 1 , w 2 ] R { 0 } is said to be harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 if
g ( x ) = g 1 1 w 1 + 1 w 2 1 x
holds for all x [ w 1 , w 2 ] .
Fejér-type fractional integral inequalities for harmonically convex functions are presented now.
Theorem 32 
([41]). Let Q : I ( 0 , ) R be a differentiable function on I , the interior of I , such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | is harmonically convex on [ w 1 , w 2 ] , g : [ w 1 , w 2 ] R is continuous and harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , then the fractional inequality is given as:
| Q 2 w 1 w 2 w 1 + w 2 J w 1 + w 2 2 w 1 w 2 + α ( Q h ) ( 1 / w 1 ) + J w 1 + w 2 2 w 1 w 2 α ( Q h ) ( 1 / w 2 ) J w 1 + w 2 2 w 1 w 2 + α ( Q g h ) ( 1 / w 1 ) + J w 1 + w 2 2 w 1 w 2 α ( Q g h ) ( 1 / w 2 ) | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α [ C 1 ( α ) | Q ( w 1 ) | + C 2 ( α ) | Q ( w 2 ) | ,
where
C 1 ( α ) = w 2 2 ( α + 1 ) ( α + 2 ) 2 F 1 ( 2 , α + 1 ; α + 3 ; 1 w 1 w 2 ) ( w 1 + w 2 ) 2 ( α + 1 ) ( α + 2 ) 2 F 1 ( 2 , α + 1 ; α + 3 ; 1 w 2 w 1 w 2 + w 1 ) , C 2 ( α ) = w 2 2 α + 2 2 F 1 ( 2 , α + 2 ; α + 3 ; 1 w 1 w 2 ) 2 ( w 1 + w 2 ) 2 α + 1 2 F 1 ( 2 , α + 1 ; α + 2 ; 1 w 2 w 1 w 2 + w 1 ) + ( w 1 + w 2 ) 2 ( α + 1 ) ( α + 2 ) 2 F 1 ( 2 , α + 1 ; α + 3 ; 1 w 2 w 1 w 2 + w 1 ) ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .
Theorem 33 
([41]). Assume that the function g is as in Theorem 32. Let Q : I ( 0 , ) R be a differentiable function on I , the interior of I , such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | q , q 1 is harmonically convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 w 2 + α ( g h ) ( 1 / w 1 ) + J 1 w 1 α ( Q h ) ( 1 / w 2 ) J 1 w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 w 1 α ( Q g h ) ( 1 / w 2 ) | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α [ C 3 1 1 q ( α ) C 4 ( α ) | Q ( w 1 ) | q + C 5 ( α ) | Q ( w 2 ) | q 1 q + C 6 1 1 q ( α ) C 7 ( α ) | Q ( w 1 ) | q + C 8 ( α ) | Q ( w 2 ) | q 1 q ] ,
where
C 3 ( α ) = ( w 1 + w 2 ) 2 2 α 1 ( α + 1 ) 2 F 1 2 , 1 , α + 2 ; w 2 w 1 w 2 + w 1 C 4 ( α ) = ( w 1 + w 2 ) 2 2 α ( α + 2 ) 2 F 1 2 , 1 , α + 3 ; w 2 w 1 w 2 + w 1 C 5 ( α ) = C 3 ( α ) C 4 ( α ) C 6 ( α ) = w 2 2 2 α + 1 ( α + 1 ) 2 F 1 2 , α + 1 ; α + 1 ; 1 2 1 w 1 w 2 , C 7 ( α ) = w 2 2 2 α + 1 ( α + 1 ) 2 F 1 2 , α + 1 ; α + 2 ; 1 2 1 w 1 w 2 w 2 2 2 α + 2 ( α + 2 ) 2 F 1 2 , α + 2 ; α + 3 ; 1 2 1 w 1 w 2 ] , C 8 ( α ) = C 6 ( α ) C 7 ( α ) ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .
Theorem 34 
([41]). Assume that the function g is as in Theorem 32. Let Q : I ( 0 , ) R be a differentiable function on I , the interior of I , such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | q , q > 1 is harmonically convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 w 2 + α ( g h ) ( 1 / w 1 ) + J 1 w 1 α ( Q h ) ( 1 / w 2 ) J 1 w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 w 1 α ( Q g h ) ( 1 / w 2 ) | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α [ C 9 1 p ( α ) | Q ( w 1 ) | q + 3 | Q ( w 2 ) | q 8 1 q + C 10 1 p ( α ) 3 | Q ( w 1 ) | q + | Q ( w 2 ) | q 8 1 q ,
where
C 9 ( α ) = ( w 1 + w 2 ) 2 p 2 α p 2 p + 1 ( α p + 1 ) 2 F 1 2 p , 1 ; α p + 2 ; w 2 w 1 w 2 + w 1 C 10 ( α ) = w 2 2 p 2 α p + 1 ( α p + 1 ) 2 F 1 2 , α p + 1 ; α p + 2 ; 1 2 1 w 1 w 2 ,
with α > 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 and 1 p + 1 q = 1 .
Theorem 35 
([42]). Let Q : I ( 0 , ) R be differentiable mapping I , where w 1 , w 2 I with w 1 < w 2 . Assume also that g : [ w 1 , w 2 ] [ 0 , ) is a continuous positive mapping symmetric to 2 w 1 w 2 w 1 + w 2 . If the mapping | Q | is harmonically-convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 / w 2 + α g h ( 1 / w 1 ) + J 1 / w 1 α g h ( 1 / w 2 ) J 1 / w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 / w 1 α ( Q g h ) ( 1 / w 2 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 ( w 1 w 2 ) α + 1 Γ ( α + 1 ) C 1 ( α ) | Q ( w 1 ) | + C 2 ( α ) | Q ( H ) | + C 3 ( α ) | Q ( w 2 ) | ,
with h ( x ) = 1 x , x 1 w 2 , 1 w 1 ,
C 1 ( α ) = 0 1 ( 1 t ) ( 1 + t ) α ( 1 t ) α ( L ( t ) ) 2 d t , C 2 ( α ) = 0 1 t ( 1 + t ) α ( 1 t ) α [ ( L ( t ) ) 2 + ( U ( t ) ) 2 ] d t , C 3 ( α ) = 0 1 ( 1 t ) ( 1 + t ) α ( 1 t ) α ( U ( t ) ) 2 d t .
and
L ( t ) = w 1 H t H + ( 1 t ) w 1 , U ( t ) = w 2 H t H + ( 1 t ) w 2 , H = H ( w 1 , w 2 ) = 2 w 1 w 2 w 1 + w 2 .
Theorem 36 
([42]). Let g be as in Theorem 35. Let Q : I ( 0 , ) R be differentiable mapping I , where w 1 , w 2 I with w 1 < w 2 and | Q | q is harmonically convex on [ w 1 , w 2 ] , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 / w 2 + α g h ( 1 / w 1 ) + J 1 / w 1 α g h ( 1 / w 2 ) J 1 / w 2 + α ( Q g h ) ( 1 / w 1 ) + J 1 / w 1 α ( Q g h ) ( 1 / w 2 ) | ( w 2 w 1 ) α + 1 g 2 α + 1 ( w 1 w 2 ) α + 1 Γ ( α + 1 ) 2 2 ( 2 α 1 ) α + 1 1 1 q × C 1 ( α , q ) | Q ( w 1 ) | q + C 2 ( α , q ) | Q ( H ) | q + C 3 ( α , q ) | Q ( w 2 ) | q 1 q ,
with h ( x ) = 1 x , x 1 w 2 , 1 w 1 ,
C 1 ( α , q ) = 0 1 ( 1 t ) ( 1 + t ) α ( 1 t ) α t ( L ( t ) ) 2 q d t , C 2 ( α , q ) = 0 1 ( 1 + t ) α ( 1 t ) α ( 1 t ) [ ( L ( t ) ) 2 q + ( U ( t ) ) 2 q ] d t , C 3 ( α , q ) = 0 1 ( 1 t ) ( 1 + t ) α ( 1 t ) α t ( U ( t ) ) 2 q d t ,
and L , U , H are given in Theorem 35.
Theorem 37 
([43]). Assume that w 1 , w 2 I with w 1 < w 2 and Q : [ w 1 , w 2 ] R is a harmonically convex function and Q L [ w 1 , w 2 ] . If g : [ w 1 , w 2 ] R is harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , integrable and non-negative then the fractional inequalities are given as:
Q 2 w 1 w 2 w 1 + w 2 J 1 w 2 + α ( g h ) 1 w 1 + J 1 w 1 α ( g h ) 1 w 2 J 1 w 2 + α ( g h ) 1 w 1 + J 1 w 1 α ( g h ) 1 w 2 Q ( w 1 ) + Q ( w 2 ) 2 J 1 w 2 + α ( g h ) 1 w 1 + J 1 w 1 α ( g h ) 1 w 2 ,
with α > 0 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .
Theorem 38 
([43]). Assume that Q is as in Theorem 37. If | Q | is harmonically convex on [ w 1 , w 2 ] and g : [ w 1 , w 2 ] R is harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 and continuous, then an inequality in the frame of the fractional operator is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 w 2 + α ( g h ) 1 w 1 + J 1 w 1 α ( g h ) 1 w 2 J 1 w 2 + α ( Q g h ) 1 w 1 + J 1 w 1 α ( Q g h ) 1 w 2 | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 w 1 w 2 α E 1 ( a ) | Q ( w 1 ) | + E 2 ( a ) | Q ( w 2 ) | ,
where
E 1 ( a ) = x 2 2 α + 2 2 F 1 2 , 1 ; α + 3 ; 1 w 1 w 2 x 2 2 ( α + 1 ) ( α + 2 ) 2 F 1 2 , α + 1 ; α + 3 ; 1 w 1 w 2 + 2 ( w 1 + w 2 ) 2 ( α + 1 ) ( α + 2 ) 2 F 1 2 , α + 1 ; α + 3 ; w 2 w 1 w 1 + w 2 , E 2 ( a ) = x 2 2 ( α + 1 ) ( α + 2 ) 2 F 1 2 , 2 ; α + 3 ; 1 w 1 w 2 x 2 2 α + 2 2 F 1 2 , α + 2 ; α + 3 ; 1 w 1 w 2 + w 1 + w 2 2 2 1 α + 1 2 F 1 2 , α + 1 ; α + 2 ; w 2 w 1 w 1 + w 2 2 ( w 1 + w 2 ) 2 ( α + 1 ) ( α + 2 ) 2 F 1 2 , α + 1 ; α + 3 ; w 2 w 1 w 1 + w 2 ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .

2.6. Fejér-Type Fractional Integral Inequalities for Harmonically Quasi-Convex Functions

Definition 7 
([44]). A function Q : I ( 0 , ) [ 0 , ) is said to be harmonically quasi-convex, if
Q x y t x + ( 1 t ) y max { Q ( x ) , Q ( y ) } ,
for all x , y I and t [ 0 , 1 ] .
Theorem 39 
([45]). Assume that Q : I ( 0 , ) R is a differentiable function on I such that Q L [ w 1 , w 2 ] , where w 1 , w 2 I and w 1 < w 2 . If | Q | is harmonically quasi-convex function on [ w 1 , w 2 ] , g : [ w 1 , w 2 ] R is continuous and harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 w 2 + α ( g h ) 1 w 1 + J 1 w 1 α ( g h ) 1 w 2 J 1 w 2 + α ( Q g h ) 1 w 1 + J 1 w 1 α ( Q g h ) 1 w 2 | g w 1 w 2 ( w 2 w 1 ) Γ ( α + 1 ) w 2 w 1 a α sup { | Q ( w 1 ) | , | Q ( w 2 ) | } [ x 2 2 α + 1 2 F 1 2 , 1 , α + 2 , 1 w 1 w 2 x 2 2 α + 1 2 F 1 2 , α + 1 , α + 2 , 1 w 1 w 2 + 4 ( w 1 + w 2 ) 2 α + 1 2 F 1 2 , α + 1 , α + 2 , 1 w 2 w 1 w 1 + w 2 ] ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 .
Theorem 40 
([45]). Assume that Q is as in Theorem 39. If | Q | q , q > 1 is harmonically quasi-convex function on [ w 1 , w 2 ] , g : [ w 1 , w 2 ] R is continuous and harmonically symmetric with respect to 2 w 1 w 2 w 1 + w 2 , then the fractional inequality is given as:
| Q ( w 1 ) + Q ( w 2 ) 2 J 1 w 2 + α ( g h ) 1 w 1 + J 1 w 1 α ( g h ) 1 w 2 J 1 w 2 + α ( Q g h ) 1 w 1 + J 1 w 1 α ( Q g h ) 1 w 2 | g w 1 w 2 ( w 2 w 1 ) 2 1 q Γ ( α + 1 ) w 2 w 1 a α sup { | Q ( w 1 ) | q , | Q ( w 2 ) | q } × [ w 1 + w 2 2 2 p 1 2 ( α p + 1 ) 2 F 1 2 p , α p + 1 , α p + 2 , 1 w 2 w 1 w 1 + w 2 + w 2 2 p 1 2 ( α p + 1 ) 2 F 1 2 p , 1 , α p + 2 , 1 2 1 w 1 w 2 ] ,
with 0 < α 1 and h ( x ) = 1 x , x 1 w 2 , 1 w 1 and 1 p + 1 q = 1 .

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
Q ( t x p + ( 1 t ) y p ) 1 / p Q ( y ) + t Φ ( Q ( x ) , Q ( y ) ) μ t ( 1 t ) ( y p x p ) 2
holds for t [ 0 , 1 ] .
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 Q : I R with magnitude μ > 0 and Φ ( . ) be bounded above in Q ( I ) × Q ( I ) and Q L [ w 1 , w 2 ] . Also, let w : [ w 1 , w 2 ] R be an integrable, non-negative and symmetric with respect to w 1 p + w 2 p 2 1 p , then
Γ ( α ) 2 Q w 1 p + w 2 p 2 1 p J w 1 p + α w g ( w 2 p ) + J w 2 p α w g ( w 1 p ) M Φ Γ ( α ) 2 J w 1 p + α w g ( w 2 p ) + J w 2 p α w g ( w 1 p ) + μ 2 w 1 p w 2 p ( 2 x w 2 p w 1 p ) 2 ( w 2 p x ) α 1 w g ( x ) d x Γ ( α ) 2 J w 1 p + α Q w g ( w 2 p ) + J w 2 p α Q w g ( w 1 p ) Q ( w 1 ) + Q ( w 2 ) 2 Γ ( α ) 2 × J w 1 p + α w g ( w 2 p ) + J w 2 p α w g ( w 1 p ) + M Φ w 2 p w 1 p w 1 p w 2 p ( w 2 p x ) α w g ( x ) d x μ w 1 p w 2 p ( w 2 p x ) 2 w g ( x ) d x .
Definition 9 
([48]). Let p R { 0 } . A function Q : [ w 1 , w 2 ] ( 0 , ) R is said to be p-symmetric with respect to w 1 p + w 2 p 2 1 p if
Q ( x ) = Q [ w 1 p + w 2 p x p ] 1 p ,
holds for all x [ w 1 , w 2 ] .
Theorem 42 
([48]). Assume that Q : I ( 0 , ) R is a p-convex function, p R { 0 } , α > 0 and w 1 , w 2 I with w 1 < w 2 . If Q L [ w 1 , w 2 ] and w : [ w 1 , w 2 ] R is non-negative, integrable and p-symmetric with respect to w 1 p + w 2 p 2 1 p , then the fractional inequalities are given as:
Q w 1 p + w 2 p 2 1 p J w 1 p + α ( w g ) ( w 2 p ) + J w 2 p α ( w g ) ( w 1 p ) J w 1 p + α ( Q w g ) ( w 2 p ) + J w 2 p α ( Q w g ) ( w 1 p ) Q ( w 1 ) + Q ( w 2 ) 2 J w 1 p + α ( w g ) ( w 2 p ) + J w 2 p α ( w g ) ( w 1 p ) , p > 0 ,
with g ( x ) = x 1 p , x [ w 1 p , w 2 p ] , and
Q w 1 p + w 2 p 2 1 p J w 2 p + α ( w g ) ( w 1 p ) + J w 1 p α ( w g ) ( w 2 p ) J w 2 p + α ( Q w g ) ( w 1 p ) + J w 1 p α ( Q w g ) ( w 2 p ) Q ( w 1 ) + Q ( w 2 ) 2 J w 2 p + α ( w g ) ( w 1 p ) + J w 1 p α ( w g ) ( w 2 p ) , p < 0 ,
with g ( x ) = x 1 p , x [ w 2 p , w 1 p ] .

2.8. Fejér-Type Fractional Integral Inequalities via Convexity with Respect to Strictly Monotone Function

Definition 10 
([38]). Let I , J be intervals in R and Q : I R be the convex function, also let σ : J I R be strictly monotone function. Then, Q is called convex with respect to σ if
Q ( σ 1 ( t x + ( 1 t ) y ) ) t Q ( σ 1 ( x ) ) + ( 1 t ) Q ( σ 1 ( y ) ) ,
for all t [ 0 , 1 ] , x , y R a n g e ( σ ) , provided R a n g e ( σ ) is a convex set.
We give a Fejér-type fractional integral inequality for convex function Q with respect to a strictly monotone function σ .
Theorem 43 
([49]). Let I , J be intervals in R and Q , Φ : [ w 1 , w 2 ] I R be real valued functions. Let Q be convex and g be positive and symmetric about σ ( w 1 ) + σ ( w 2 ) 2 . Let σ : J [ w 1 , w 2 ] R be a strictly monotone function. If Q is convex with respect to σ , then the fractional inequality is given as:
Q σ 1 σ ( w 1 ) + σ ( w 2 ) 2 J σ ( w 1 ) + α Φ ( w 2 ) + J σ ( w 2 ) α Φ ( w 1 ) J σ ( w 1 ) + α ( Q Φ ) ( w 2 ) + J σ ( w 2 ) α ( Q Φ ) ( w 1 ) Q ( w 1 ) + Q ( w 2 ) 2 J σ ( w 1 ) + α Φ ( w 2 ) + J σ ( w 2 ) α Φ ( w 1 ) .
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:
Q σ 1 σ ( w 1 ) + σ ( w 2 ) 2 J σ ( w 1 ) + σ ( w 2 ) 2 + α Φ ( w 2 ) + J σ ( w 1 ) + σ ( w 2 ) 2 α Φ ( w 1 ) J σ ( w 1 ) + σ ( w 2 ) 2 + α ( Q Φ ) ( w 2 ) + J σ ( w 1 ) + σ ( w 2 ) 2 α ( Q Φ ) ( w 1 ) Q ( w 1 ) + Q ( w 2 ) 2 J σ ( w 1 ) + σ ( w 2 ) 2 + α Φ ( w 2 ) + J σ ( w 1 ) + σ ( w 2 ) 2 α Φ ( w 1 ) .

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 Q : Δ = [ w 1 , w 2 ] × [ z 1 , z 2 ] R will be called co-ordinated convex on Δ, for all t , s [ 0 , 1 ] and ( x , y ) , ( u , w ) Δ , if the following inequality holds:
Q ( t x + ( 1 t ) y , s u + ( 1 s ) w ) t s Q ( x , u ) + s ( 1 t ) Q ( y , u ) + t ( 1 s ) Q ( x , w ) + ( 1 t ) ( 1 s ) Q ( y , w ) .
Definition 12 
([51]). Assume that Q L 1 ( [ w 1 , w 2 ] × [ z 1 , z 2 ] ) . The Riemann–Liouville integrals J w 1 + , z 1 + α , β Q , J w 1 + , z 2 α , β Q , J w 2 , z 1 + α , β Q , and J w 2 , z 2 α , β Q of order α , β > 0 with a , c 0 are defined by
J w 1 + , z 1 + α , β Q ( x , y ) = 1 Γ ( α ) Γ ( β ) w 1 x z 1 y ( x t ) α 1 ( y s ) β 1 Q ( s , t ) d s d t , x > w 1 , y > z 1 , J w 1 + , z 2 α , β Q ( x , y ) = 1 Γ ( α ) Γ ( β ) w 1 x y z 2 ( x t ) α 1 ( s y ) β 1 Q ( s , t ) d s d t , x > w 1 , y < z 2 , J w 2 , z 1 + α , β Q ( x , y ) = 1 Γ ( α ) Γ ( β ) x w 2 z 1 y ( t x ) α 1 ( y s ) β 1 Q ( s , t ) d s d t , x < w 2 , y > z 1 , J w 2 , z 2 α , β Q ( x , y ) = 1 Γ ( α ) Γ ( β ) x w 2 y z 2 ( t x ) α 1 ( s y ) β 1 Q ( s , t ) d s d t , x < w 2 , y < z 2 ,
respectively.
Theorem 45 
([51]). Let Q : Δ R 2 R be a co-ordinated convex on Δ = [ w 1 , w 2 ] × [ z 1 , z 2 ] in R 2 with 0 w 1 < w 2 , 0 z 1 < z 2 and Q L 1 ( Δ ) . If g : Δ R is non-negative, integrable and symmetric to w 1 + w 2 2 and z 1 + z 2 2 , then for all α , β > 0 and λ , γ [ 0 , 1 ] , we have the inequalities:
Q λ w 2 + ( 2 λ ) w 1 2 , γ z 2 + ( 2 γ ) z 1 2 J w 2 , z 2 α , β ( g ) ( λ w 1 + ( 1 λ ) w 2 , γ z 1 + ( 1 γ ) z 2 ) J w 1 + , z 1 + α , β ( Q g ) ( w 1 + λ ( w 2 w 1 ) , z 1 + γ ( z 2 z 1 ) ) + J w 1 + , z 2 α , β ( Q g ) ( w 1 + λ ( w 2 w 1 ) , γ z 1 + ( 1 γ ) z 2 ) + J w 2 , z 1 + α , β ( Q g ) ( λ w 1 + ( 1 λ ) w 2 , z 1 + γ ( z 2 z 1 ) ) + J w 2 , z 2 α , β ( Q g ) ( λ w 1 + ( 1 λ ) w 2 , γ z 1 + ( 1 γ ) z 2 ) J w 2 , z 2 α , β ( g ) ( λ w 1 + ( 1 λ ) w 2 , γ z 1 + ( 1 γ ) z 2 ) Q ( w 1 , z 1 ) + Q ( w 2 , z 1 ) + Q ( w 1 , z 2 ) + Q ( w 2 , z 2 ) 4 .
Theorem 46 
([51]). Assume that the conditions of Theorem 45 are satisfied. Then, we have the inequalities:
J w 1 + , [ z 2 γ ( z 2 z 1 ) ] + α , β Q w 2 , γ z 2 + ( 2 γ ) z 1 2 g ( w 2 , z 2 ) + J w 1 + , z 2 α , β Q w 2 , γ z 2 + ( 2 γ ) z 1 2 g ( w 2 , z 2 γ ( z 2 z 1 ) ) + J w 2 , [ z 2 γ ( z 2 z 1 ) ] + α , β Q w 1 , γ z 2 + ( 2 γ ) z 1 2 g ( w 1 , z 2 γ ( z 2 z 1 ) ) + J w 2 , z 2 α , β Q w 1 , γ z 2 + ( 2 γ ) z 1 2 g ( w 1 , z 2 γ ( z 2 z 1 ) ) + J z 1 + , [ w 2 λ ( w 2 w 1 ) ] + β , α Q λ w 2 + ( 2 λ ) w 1 2 , z 2 g ( w 2 , z 2 ) + J z 1 + , w 2 β , α Q λ w 2 + ( 2 λ ) w 1 2 , z 2 g ( w 2 λ ( w 2 w 1 ) , z 2 ) + J z 2 , [ w 2 λ ( w 2 w 1 ) ] + β , α Q λ w 2 + ( 2 λ ) w 1 2 , z 1 g ( w 2 , z 1 ) + J z 2 , w 2 β , α Q λ w 2 + ( 2 λ ) w 1 2 , z 1 g ( w 2 , z 1 ) J w 1 + , z 1 + α , β ( Q g ) ( w 2 , z 1 + γ ( z 2 z 1 ) ) + J w 1 + , z 2 α , β ( Q g ) ( w 2 , z 2 γ ( z 2 z 1 ) ) + J w 2 , z 1 + α , β ( Q g ) ( w 1 , z 1 + γ ( z 2 z 1 ) ) + J w 2 , z 2 + α , β ( Q g ) ( w 1 , z 2 γ ( z 2 z 1 ) ) + J z 1 + , w 1 + β , α ( Q g ) ( w 1 + λ ( w 2 w 1 ) , z 2 ) + J z 1 + , w 2 β , α ( Q g ) ( z 2 , w 2 λ ( w 2 w 1 ) ) + J z 2 , w 1 + β , α ( Q g ) ( w 1 + λ ( w 2 w 1 ) , z 1 ) + J z 2 , w 2 β , α ( Q g ) ( w 2 λ ( w 2 w 1 ) , z 1 ) 1 2 J w 1 + , z 1 + α , β [ Q ( w 2 , z 1 ) + Q ( w 2 , z 2 ) ] g ( w 2 , z 1 + γ ( z 2 z 1 ) ) + 1 2 J w 1 + , z 2 α , β [ Q ( w 2 , z 1 ) + Q ( w 2 , z 2 ) ] g ( w 2 , z 2 γ ( z 2 z 1 ) ) + 1 2 J w 2 , z 1 + α , β [ Q ( w 1 , z 1 ) + Q ( w 1 , z 2 ) ] g ( w 1 , z 1 + γ ( z 2 z 1 ) ) + 1 2 J w 2 , z 2 α , β [ Q ( w 1 , z 1 ) + Q ( w 1 , z 2 ) ] g ( w 1 , z 2 γ ( z 2 z 1 ) ) + 1 2 J z 1 + , w 1 + β , α [ Q ( w 1 , z 2 ) + Q ( w 2 , z 2 ) ] ( Q g ) ( w 1 + λ ( w 2 w 1 ) , z 2 ) + 1 2 J z 1 + , w 2 β , α [ Q ( w 1 , z 2 ) + Q ( w 2 , z 2 ) ] ( Q g ) ( w 2 + λ ( w 2 w 1 ) , z 2 ) + 1 2 J z 2 , w 1 + β , α [ Q ( w 1 , z 1 ) + Q ( w 2 , z 1 ) ] ( Q g ) ( w 1 + λ ( w 2 w 1 ) , z 1 ) + 1 2 J z 2 , w 2 β , α [ Q ( w 1 , z 1 ) + Q ( w 2 , z 1 ) ] ( Q g ) ( w 2 λ ( w 2 w 1 ) , z 1 ) .
Theorem 47 
([52]). Let Q : Δ R 2 R be co-ordinated convex on Δ = [ w 1 , w 2 ] × [ z 1 , z 2 ] in R 2 with 0 w 1 < w 2 , 0 z 1 < z 2 and Q L 1 ( Δ ) . If g : Δ R is non-negative, integrable, and symmetric to w 1 + w 2 2 and z 1 + z 2 2 , then for all α , β > 0 , we have the inequalities:
Q w 1 + w 2 2 , z 1 + z 2 2 [ J w 1 + , z 1 + α , β g ( w 2 , z 2 ) + J w 1 + , z 2 α , β g ( w 2 , z 1 ) + J w 2 , z 1 + α , β g ( w 1 , z 2 ) + J w 2 , z 2 α , β g ( w 1 , z 1 ) ] 1 4 [ J w 1 + , z 1 + α , β ( Q g ) ( w 2 , z 2 ) + J w 1 + , z 2 α , β ( Q g ) ( w 2 , z 1 ) + J w 2 , z 1 + α , β ( Q g ) ( w 1 , z 2 ) + J w 2 , z 2 α , β ( Q g ) ( w 1 , z 1 ) ] Q ( w 1 , z 1 ) + Q ( w 1 , z 2 ) + Q ( w 2 , z 1 ) + Q ( w 2 , z 2 ) 4 × J w 1 + , z 1 + α , β g ( w 2 , z 2 ) + J w 1 + , z 2 α , β g ( w 2 , z 1 ) + J w 2 , z 1 + α , β g ( w 1 , z 2 ) + J w 2 , z 2 α , β g ( w 1 , z 1 ) .
Theorem 48 
([52]). Under the conditions of Theorem 47, we have the fractional inequalities:
Q w 1 + w 2 2 , z 1 + z 2 2 [ J w 1 + , z 1 + α , β g ( w 2 , z 2 ) + J w 1 + , z 2 α , β g ( w 2 , z 1 ) + J w 2 , z 1 + α , β g ( w 1 , z 2 ) + J w 2 , z 2 α , β g ( w 1 , z 1 ) ] J w 1 + α Q w 2 , z 1 + z 2 2 J z 1 + β g ( w 2 , z 2 ) + J w 1 + α Q w 2 , z 1 + z 2 2 J z 2 β g ( w 2 , z 1 ) + J w 2 α Q w 1 , z 1 + z 2 2 J z 1 + β g ( w 1 , z 2 ) + J w 2 α Q w 1 , z 1 + z 2 2 J z 2 β g ( w 1 , z 1 ) + J z 1 + β Q w 1 + w 2 2 , z 2 J w 1 + α g ( w 2 , z 2 ) + J z 1 + β Q w 1 + w 2 2 , z 2 J w 2 α g ( w 1 , z 2 ) + J z 2 β Q w 1 + w 2 2 , z 1 J w 1 + α g ( w 2 , z 1 ) + J z 2 β Q w 1 + w 2 2 , z 1 J w 2 α g ( w 1 , z 1 ) 2 [ J w 1 + , z 1 + α , β ( Q g ) ( w 2 , z 2 ) + J w 1 + , z 2 α , β ( Q g ) ( w 2 , z 1 ) + J w 2 , z 1 + α , β ( Q g ) ( w 1 , z 2 ) + J w 2 , z 2 α , β ( Q g ) ( w 1 , z 1 ) ] J w 1 + α Q ( w 2 , z 1 ) J z 1 + β g ( w 2 , z 2 ) + J w 1 + α Q ( w 2 , z 2 ) J z 2 β g ( w 2 , z 1 ) + J w 2 α Q ( w 1 , z 1 ) J z 1 + β g ( w 1 , z 2 ) + J w 2 α Q ( w 1 , z 2 ) J z 2 β g ( w 1 , z 1 ) + J z 1 + β Q ( w 1 , z 2 ) J w 1 + α g ( w 2 , z 2 ) + J z 1 + β Q ( w 2 , z 2 ) J z 2 α g ( w 1 , z 1 ) + J z 2 β Q ( w 1 , z 1 ) J w 1 + α g ( w 2 , z 1 ) + J z 2 β Q ( w 2 , z 1 ) J z 2 α g ( w 1 , z 1 ) Q ( w 1 , z 1 ) + Q ( w 1 , z 2 ) + Q ( w 2 , z 1 ) + Q ( w 2 , z 2 ) 4 × J w 1 + , z 1 + α , β g ( w 2 , z 2 ) + J w 1 + , z 2 α , β g ( w 2 , z 1 ) + J w 2 , z 1 + α , β g ( w 1 , z 2 ) + J w 2 , z 2 α , β g ( w 1 , z 1 ) .

2.10. Fejér-Type Fractional Integral Inequalities for ( θ , h m ) p -Convex Functions

Definition 13 
([53]). Let J R be an interval containing ( 0 , 1 ) and let h : J R be a non-negative function. Let I ( 0 , ) be an interval and p R { 0 } . A function Q : I R is said to be ( θ , h m ) p -convex, if
Q t w 1 p + m ( 1 t ) w 2 p 1 p h ( t θ ) Q ( w 1 ) + m h ( 1 t θ ) Q ( w 2 ) ,
holds, provided that t w 1 p + m ( 1 t ) w 2 p 1 p I for t [ 0 , 1 ] and ( θ , m ) [ 0 , 1 ] 2 .
Theorem 49 
([53]). Let Q : I R be a positive ( θ , h m ) p -convex function with t w 2 p + m ( 1 t ) ( w 1 p / m ) 1 p I , m 0 , w 1 p < m w 2 p . If F : I R is a positive function, then
Q w 1 p + m w 1 p 2 1 / p I w 1 p + τ F ξ m w 1 p h 1 2 α I w 1 p + τ Q F ξ m w 2 p + m μ + 1 h 2 α 1 2 α I w 2 p τ Q F ξ w 1 p m m w 2 p w 1 p τ Γ ( τ ) h 1 2 α Q ( w 1 ) + m h 2 α 1 2 α Q ( w 2 ) × 0 1 t τ 1 F t w 1 p + m ( 1 t ) w 2 p 1 p h t α d t + m h 1 2 α Q ( w 2 ) + m h 2 α 1 2 α Q w 1 m 2 × 0 1 t τ 1 F t w 1 p + m ( 1 t ) w 2 p 1 / p h 1 t α d t , ξ ( t ) = t 1 / p , Q F ξ = ( Q ξ ) ( F ξ ) .
Theorem 50 
([53]). Assume that Q is as in Theorem 49. Then, the following inequalities hold:
Q w 1 p + m w 2 p 2 1 / p I w 1 p + m w 2 p / 2 + τ F ξ m w 2 p h 1 2 α I w 1 p + m w 2 p / 2 + τ Q F ξ m w 2 p + m μ + 1 h 2 α 1 2 α I w 1 p + m w 2 w 2 p / 2 τ Q F ξ w 1 p m 1 Γ ( τ ) m w 2 p w 1 p 2 τ h 1 2 α Q ( w 1 ) + m h 2 α 1 2 α Q ( w 2 ) × 0 1 t τ 1 F t 2 w 1 p + m 2 t 2 w 2 p 1 / p h t 2 α d t + m h 1 2 α Q ( w 2 ) + m h 2 α 1 2 α Q w 1 m 2 × 0 1 t τ 1 F t 2 w 1 p + m 2 t 2 w 2 p 1 / p h 2 α t α 2 α d t , ξ ( t ) = t 1 / p , Q F ξ = ( Q ξ ) ( F ξ ) .

2.11. Fejér-Type Fractional Integral Inequalities for h-Preinvex Functions

Definition 14 
([54]). Let Q : X R and Φ : X × X R n , where X is a non-empty closed set in R n , be continuous function. Assume that h : [ 0 , 1 ] R . Then, Q is said to be h-preinvex with respect to Φ , if
Q ( w 1 + Φ ( w 2 , w 1 ) ) h ( 1 ) Q ( w 1 ) + h ( ) Q ( w 2 ) ,
for all w 1 , w 2 X and [ 0 , 1 ] , where Q ( · ) > 0 .
Theorem 51 
([55]). Let Q : [ w 1 , w 1 + Φ ( w 2 , w 1 ) ] R be a h-preinvex function, condition C for Φ holds, and Φ ( w 2 , w 1 ) > 0 , h ( 1 2 ) > 0 and F : A [ 0 , ) be differentiable and symmetric to w 1 + 1 2 Φ ( w 2 , w 1 ) , then we have
Γ ( α ) 2 · h 1 2 · Φ ( w 2 , w 1 ) α Q w 1 + 1 2 Φ ( w 2 , w 1 ) I ( w 1 + Φ ( w 2 , w 1 ) ) α F ( w 1 ) + I w 1 + α F ( w 1 + Φ ( w 2 , w 1 ) ) Γ ( α ) Φ ( w 2 , w 1 ) α I ( w 1 + Φ ( w 2 , w 1 ) ) α α F ( w 1 ) Q ( w 1 ) + I w 1 + α F ( w 1 + Φ ( w 2 , w 1 ) ) Q ( w 1 + Φ ( w 2 , w 1 ) ) [ Q ( w 1 ) + Q ( w 2 ) ] · 0 1 α 1 [ h ( ) + h ( 1 ) ] F ( w 1 + Φ ( w 2 , w 1 ) ) d .
Theorem 52 
([55]). Let A R be an open invex subset with respect to Φ : A × A R and w 1 , w 2 A with Φ ( w 2 , w 1 ) > 0 . Suppose that Q : A R is a differentiable mapping on A and F : A [ 0 , ) is differentiable and symmetric to w 1 + 1 2 Φ ( w 2 , w 1 ) . If | Q | is h-preinvex on A, we have
| Γ ( α + 1 ) Φ ( w 2 , w 1 ) α + 1 I w 1 + α F ( w 1 + Φ ( w 2 , w 1 ) ) Q ( w 1 + Φ ( w 2 , w 1 ) ) + I ( w 1 + Φ ( w 2 , w 1 ) ) α F ( w 1 ) Q ( w 1 ) I w 1 + α + 1 F ( w 1 + Φ ( w 2 , w 1 ) ) Q ( w 1 + Φ ( w 2 , w 1 ) ) + I ( w 1 + Φ ( w 2 , w 1 ) ) α + 1 F ( w 1 ) Q ( w 1 ) 1 Φ ( w 2 , w 1 ) [ Q ( w 1 + Φ ( w 2 , w 1 ) ) F ( w 1 + Φ ( w 2 , w 1 ) ) + Q ( w 1 ) F ( w 1 ) ] | Q ( w 1 ) + Q ( w 2 ) · 0 1 α F ( w 1 + Φ ( w 2 , w 1 ) ) [ h ( ) + h ( 1 ) ] d .
Theorem 53 
([55]). Let A R be an open invex subset with respect to Φ : A × A R and w 1 , w 2 A with Φ ( w 2 , w 1 ) > 0 . Suppose that Q : A R is a differentiable mapping on A and F : A [ 0 , ) is differentiable and symmetric to w 1 + 1 2 Φ ( w 2 , w 1 ) . If | Q | is h-preinvex on A and q 1 , we have
| Γ ( α + 1 ) Φ ( w 2 , w 1 ) α + 1 I w 1 + α F ( w 1 + Φ ( w 2 , w 1 ) ) Q ( w 1 + Φ ( w 2 , w 1 ) ) + I ( w 1 + Φ ( w 2 , w 1 ) ) α F ( w 1 ) Q ( w 1 ) I w 1 + α + 1 F ( w 1 + Φ ( w 2 , w 1 ) ) Q ( w 1 + Φ ( w 2 , w 1 ) ) + I ( w 1 + Φ ( w 2 , w 1 ) ) α + 1 F ( w 1 ) Q ( w 1 ) 1 Φ ( w 2 , w 1 ) [ Q ( <