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Review

Ostrowski-Type Fractional Integral Inequalities: A Survey

1
Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
2
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
3
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
*
Author to whom correspondence should be addressed.
Foundations 2023, 3(4), 660-723; https://doi.org/10.3390/foundations3040040
Submission received: 14 October 2023 / Revised: 7 November 2023 / Accepted: 9 November 2023 / Published: 13 November 2023

Abstract

:
This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, ( ζ , m ) -convex functions, s-convex functions, ( s , r ) -convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, M T -convex functions, P-convex functions, m-convex functions, ( s , m ) -convex functions, exponentially s-convex functions, ( β , m ) -convex functions, exponential-convex functions, ζ ¯ , β , γ , δ -convex functions, quasi-geometrically convex functions, s e -convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented.

1. Introduction

The theory of convex analysis offers robust ideas and methodologies to address an extensive spectrum of issues in applied sciences. Numerous mathematicians and researchers have been striving to implement innovative ideas of convexity theory to handle the real world problems arising in nonlinear programming, statistics, control theory, optimization, etc. The theory of convexity also plays a leading role in establishing a wide class of inequalities. The theory of inequalities in the framework of fractional operators gives rise to integral inequalities. The Ostrowski-type inequalities have been developed in the literature for various types of convex functions. Ostrowski derived the following remarkable and amazing integral inequality in 1938.
Theorem 1
([1]). Let Π : I R be a differentiable function in the interior I of I , and let ς 1 , ς 2 I with ς 1 < ς 2 . If | Π ( x ) | M for all x [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( t ) d t | M ( ς 2 ς 1 ) 1 4 + x ς 1 + ς 2 2 2 ( ς 2 ς 1 ) 2
x [ ς 1 , ς 2 ] . In the above famous integral inequality the constant value 1 4 is an amazing choice in the aspect that it cannot be substituted by a smaller one.
The Ostrowski-type inequality is found to be an exalted and applicable tool in several branches of mathematics. Integral inequalities, which are used to determine the bounds of physical quantities, find extensive applications in operator theory, statistics, probability theory, numerical integration, nonlinear analysis, information theory, stochastic analysis, approximation theory, biological sciences, physics and technology. Many researchers have shown a keen interest in developing several variants and aspects of this inequality.
During the past few decades, fractional calculus has evolved as a fast-emerging and prominent area of investigation due to the nonlocal nature of fractional order integral and derivative operators. The tools of fractional calculus have been widely applied to formulate the mathematical models associated with various phenomena and processes occurring in engineering and scientific disciplines. The importance and applications of fractional calculus are eminent in the related literature. In the realm of inequalities, fractional order operators have played a fundamental role in the advancement of the topic. In particular, fractional integral operators are found to be of exceptional value in generalizing the standard integral inequalities. Here, we recall that certain inequalities are quite helpful in investigating optimization problems.
The main aim of this manuscript is to present an up-to-date review of Ostrowski-type inequalities involving different convexities and fractional integral operators. In each section/subsection of the paper, we first describe the fractional integral operators and convexities, related to the results collected for Ostrowski-type fractional integral inequalities. We provide comprehensive details for each Ostrowski-type inequality collected in this survey (without proof) for the convenience of the reader. Our survey paper contains the state-of-the-art literature review on fractional Ostrowski-type inequalities and serves as an excellent platform for the researchers who wish to initiate/develop new work on such inequalities.
The structure of this review paper is designed as follows. Section 2 summarizes Ostrowski-type fractional integral inequalities for different families of convexities, including classical convex functions, quasi-convex functions, ( ζ , m ) -convex functions, s-convex functions, ( s , r ) -convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, M T -convex functions, P-convex, m-convex functions, ( s , m ) -convex functions, exponentially s-convex functions, ( β , m ) -convex functions, exponential-convex functions, ζ ¯ , β , γ , δ -convex functions, quasi-geometrically convex functions, s e -convex functions and n-polynomial exponentially s-convex functions. Section 3 consists of Ostrowski-type fractional integral inequalities for Katugampola fractional integral operators. In Section 4, we present Ostrowski-type fractional integral inequalities involving k-Riemann–Liouville fractional integrals. Section 5 is concerned with Ostrowski-type fractional integral inequalities for preinvex functions, while Ostrowski-type fractional integral inequalities involving fractional integrals with respect to another function are described in Section 6. Mercer-Ostrowski-type fractional integral inequalities for Riemann–Liouville fractional integral operators are included in Section 7. Ostrowski-type fractional integral inequalities obtained via Hadamard fractional integral are discussed in Section 8. We collect Ostrowski-type fractional integral inequalities for integrals with exponential kernel function in Section 9. Section 10 deals with Ostrowski-type fractional integral inequalities for Atangana-Baleanu-type fractional integral operators, while Section 11 contains Ostrowski-type inequalities in terms of generalized fractional integral operators. In Section 12, we discuss Ostrowski-type fractional integral inequalities obtained via operators of quantum-calculus and Ostrowski-type inequalities of tensorial type are presented in Section 13.

2. Ostrowski-Type Inequalities via Riemann–Liouville Fractional Integral

First, we add the definitions of fractional operators, namely Riemann–Liouville, in the left and right aspects.
Definition 1
([2]). Let Π L [ ς 1 , ς 2 ] . Then, the Riemann–Liouville integrals (left and right aspect) J ς 1 + ζ Π and J ς 2 ζ Π , ζ > 0 , ς 1 0 are stated by
J ς 1 + ζ Π ( x ) = 1 Γ ( ζ ) ς 1 x ( x t ) ζ 1 Π ( t ) d t , x > ς 1 ,
and
J ς 2 ζ Π ( x ) = 1 Γ ( ζ ) x ς 2 ( t x ) ζ 1 Π ( t ) d t , x < ς 2 ,
respectively. Here, Γ ( ζ ) represent the Euler Gamma function and J ς 1 + 0 Π ( x ) = J ς 2 0 Π ( x ) = Π ( x ) .

2.1. Ostrowski-Type Fractional Integral Inequalities for Functions with Bounded Derivative

In this subsection, we present results on Ostrowski-type fractional integral inequalities for functions with bounded derivatives. We have the following results that provide lower and upper bounds for the Ostrowski differences.
Theorem 2
([3]). Let Π : [ ς 1 , ς 2 ] R be a differentiable mapping on ( ς 1 , ς 2 ) and | Π ( x ) | M for all x [ ς 1 , ς 2 ] . Then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) M ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ζ + 1 ,
for all x [ ς 1 , ς 2 ] and ζ 0 .
Theorem 3
([3]). Let the assumptions of this theorem be as stated in Theorem 2 and p > 1 with 1 p + 1 q = 1 . Then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) 1 ( ζ q + 1 ) 1 q ( x ς 1 ) ζ + 1 q Π p , [ ς 1 , x ] + ( ς 2 x ) ζ + 1 q Π p , [ x , ς 2 ] ,
x [ ς 1 , ς 2 ] and ζ 0 where Π p , [ ς 1 , x ] = ς 1 x | Π ( y ) | p d y 1 p .
Theorem 4
([4]). Let the assumptions of this theorem be as stated in Theorem 2. Then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ Γ ( ζ + 1 ) Π ( x ) J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 Γ ( ζ + 2 ) ,
for all x [ ς 1 , ς 2 ] and ζ 0 .
Theorem 5
([4]). Let the assumptions of this theorem be as stated in Theorem 2 and Π L 2 [ ς 1 , ς 2 ] . If m | Π ( x ) | M for all x [ ς 1 , ς 2 ] , then
| ζ Π ( x ) + Π ( ς 1 ) Γ ( ζ ) ( ζ + 1 ) ( x ς 1 ) ζ 1 ζ x ς 1 J x ζ Π ( ς 1 ) + ζ Π ( x ) + Π ( ς 2 ) Γ ( ζ ) ( ζ + 1 ) ( ς 2 x ) ζ 1 ζ ς 2 x J x ζ Π ( ς 2 ) | 1 2 ζ + 1 1 ( ζ + 1 ) 2 ( x ς 1 ) ζ K 1 + ( ς 2 x ) ζ K 2 Γ ( ζ ) 1 2 ζ + 1 1 ( ζ + 1 ) 2 ( x ς 1 ) ζ + ( ς 2 x ) ζ 2 Γ ( ζ ) ( M m )
for all x [ ς 1 , ς 2 ] and ζ 1 , where
K 1 2 = M ( Π 2 ; ς 1 , x ) M 2 ( Π ; ς 1 , x ) , K 2 2 = M ( Π 2 ; x ς 2 ) M 2 ( Π ; x , ς 2 )
and M ( Π ; ς 1 , ς 2 ) = 1 ς 2 ς 1 ς 1 ς 2 Π ( x ) d x .
In the next result, fractional integral inequalities of Ostrowski-Grüss type are presented.
Theorem 6
([5]). Suppose Π : I R is a differentiable function and ς 1 , ς 2 I with ς 1 < ς 2 . If Π : ( ς 1 , ς 2 ) R is bounded on ( ς 1 , ς 2 ) with m Π ( x ) M , for all x [ ς 1 , ς 2 ] , then
| Π ( x ) Γ ( ζ ) ( ς 2 x ) 1 ζ ς 2 ς 1 J ς 1 ζ Π ( ς 2 ) + ( ς 2 x ) 1 ζ J ς 1 ζ 1 Π ( ς 2 ) Π ( ς 2 ) Π ( ς 1 ) ς 2 ς 1 ( ς 2 x ) 1 ζ ( ς 2 ς 1 ) ζ Γ ( ζ + 2 ) ς 2 x Γ ( ζ + 1 ) | ( ς 2 ς 1 ) ( K ( x ) ) 1 2 1 ( ς 2 ς 1 ) Γ 2 ( ζ ) Π 2 2 Π ( ς 2 ) Π ( ς 1 ) ( ς 2 ς 1 ) Γ ( ζ ) 2 1 2 ( K ( x ) ) 1 2 2 Γ ( ζ ) ( ς 2 ς 1 ) ( M m ) ,
for all x [ ς 1 , ς 2 ] and ζ 1 , where
K ( x ) = ( ς 2 x ) 1 ζ ( ς 2 ς 1 ) 2 ζ 2 1 2 ζ + 1 + 1 2 ζ 1 1 ζ + ( v 2 x ) ζ ( ς 2 ς 1 ) 2 ς 2 x ζ ς 2 ς 1 2 ζ 1 ( ς 2 x ) 1 ζ ( ς 2 ς 1 ) ζ 1 ζ ( ζ + 1 ) ς 2 x ζ ( ς 2 ς 1 ) 2 .
Now we give one more Ostrowski-Grüss-type inequality of fractional type.
Theorem 7
([6]). Let the assumptions of this theorem be stated in Theorem 2. Then
| 1 2 Π ( x ) ( ζ + 1 ) Γ ( ζ ) ( ς 2 x ) 1 ζ 2 ( ς 2 ς 1 ) J ς 1 ζ Π ( ς 2 ) + 1 2 ( ς 2 x ) 1 ζ Γ ( ζ ) J ς 1 ζ 1 Π ( ς 2 ) + ( ς 2 x ) 2 ζ 2 ( ς 2 ς 1 ) Γ ( ζ ) J ς 1 ζ 1 Π ( ς 2 ) + ( ς 2 x ) 1 ζ ( x ς 1 ) 2 ( ς 2 ς 1 ) 2 ζ Π ( ς 1 ) | M ( ς 2 x ) 1 ζ ς 2 ς 1 ( ς 2 ς 1 ) ζ ( x ς 1 ) + ( ς 2 x ) ζ ( ς 1 + ς 1 2 x ) 2 ζ ,
where ς 1 x < ς 2 .

2.2. Ostrowski-Type Fractional Integral Inequalities for Convex Functions

Definition 2
([7]). A real-valued function Π is convex on an interval I, if
Π λ ς 1 + 1 λ ς 2 λ Π ς 1 + 1 λ Π ς 2 ,
holds for all ς 1 , ς 2 I and λ [ 0 , 1 ] .
In the following theorems, we show the Ostrowski-type inequalities in the frame of Riemann–Liouville fractional integrals for absolutely continuous and convex functions.
Theorem 8
([8]). Let Π : [ ς 1 , ς 2 ] R be an absolutely continuous function on [ ς 1 , ς 2 ] . If x ( ς 1 , ς 2 ) and there exist real numbers m 1 ( x ) , M 1 ( x ) , m 2 ( x ) , M 2 ( x ) such that
m 1 ( x ) Π ( t ) M 1 ( x ) , f o r   a l l t ( ς 1 , x )
and
m 2 ( x ) Π ( t ) M 2 ( x ) , f o r   a l l t ( x , ς 2 ) .
Then
1 Γ ( ζ + 2 ) m 2 ( x ) ( ς 2 x ) ζ + 1 M 1 ( x ) ( x ς 1 ) ζ + 1 1 Γ ( ζ + 1 ) ( x ς 1 ) ζ Π ( ς 1 ) + ( ς 2 x ) ζ Π ( ς 2 ) J ς 1 + ζ Π ( x ) J ς 2 ζ Π ( x ) Γ ( ζ + 2 ) M 2 ( x ) ( ς 2 x ) ζ + 1 m 1 ( x ) ( x ς 1 ) ζ + 1
and
1 Γ ( ζ + 2 ) m 2 ( x ) ( ς 2 x ) ζ + 1 M 1 ( x ) ( x ς 1 ) ζ + 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) 1 Γ ( ζ + 1 ) ( x ς 1 ) ζ Π ( x ) + ( ς 2 x ) ζ Π ( x ) 1 Γ ( ζ + 2 ) M 2 ( x ) ( ς 2 x ) ζ + 1 m 1 ( x ) ( x ς 1 ) ζ + 1 .
Theorem 9
([8]). Let Π : [ ς 1 , ς 2 ] R be a convex function and x ( ς 1 , ς 2 ) . Then
1 Γ ( ζ + 2 ) Π + ( x ) ( ς 2 x ) ζ + 1 Π ( x ) ( x ς 1 ) ζ + 1 1 Γ ( ζ + 1 ) ( x ς 1 ) ζ Π ( ς 1 ) + ( ς 2 x ) ζ Π ( ς 2 ) J ς 1 + ζ Π ( x ) J ς 2 ζ Π ( x ) 1 Γ ( ζ + 2 ) Π ( ς 2 ) ( ς 2 x ) ζ + 1 Π ( ς 1 ) ( x ς 1 ) ζ + 1
and
1 Γ ( ζ + 2 ) Π + ( x ) ( ς 2 x ) ζ + 1 Π ( x ) ( x ς 1 ) ζ + 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) 1 Γ ( ζ + 1 ) ( x ς 1 ) ζ Π ( x ) + ( ς 2 x ) ζ Π ( x ) 1 Γ ( ζ + 2 ) Π ( ς 2 ) ( ς 2 x ) ζ + 1 Π + ( ς 1 ) ( x ς 1 ) ζ + 1 ,
where Π ± ( · ) are the lateral derivatives of Π .
Theorem 10
([9]). Let Π : [ ς 1 , ς 2 ] R be a function which is differentiable on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | is convex on [ ς 1 , ς 2 ] and x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) + J x ζ Π ( ς 1 ) | 1 ζ + 2 { ( ς 2 x ) ζ + 2 ( ς 2 ς 1 ) ζ + 2 + ( x ς 1 ) ζ + 2 ( ς 2 ς 1 ) ζ + 2 1 ζ + 1 + ς 2 x ς 2 ς 1 | Π ( ς 1 ) | + ( x ς 1 ) ζ + 2 ( ς 2 ς 1 ) ζ + 2 + ( ς 2 x ) ζ + 2 ( ς 2 ς 1 ) ζ + 2 1 ζ + 1 + x ς 1 ς 2 ς 1 | Π ( ς 2 ) | } .
Theorem 11
([9]). Let Π be as in Theorem 10. If | Π | q , q > 1 is convex on [ ς 1 , ς 2 ] , and x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) + J x ζ Π ( ς 1 ) | 1 ( ς 2 ς 1 ) ζ + 1 ( ζ p + 1 ) 1 p [ ( ς 2 x ) ζ + 1 | Π ( ς 1 ) | q + | Π ( ς 2 ) | q 2 1 q + ( x ς 1 ) ζ + 1 | Π ( ς 1 ) | q + | Π ( ς 2 ) | q 2 1 q ] ,
where 1 p + 1 q = 1 and ζ > 0 .
Theorem 12
([9]). Let Π be as in Theorem 10. If | Π | q , q 1 is convex on [ ς 1 , ς 2 ] , and x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) + J x ζ Π ( ς 1 ) | 1 ζ + 1 1 1 q 1 ζ + 2 1 q × { ς 2 x ς 2 ς 1 ζ + 1 ς 2 x ς 2 ς 1 | Π ( ς 1 ) | q + 1 ζ + 1 + x ς 1 ς 2 ς 1 | Π ( ς 2 ) | q 1 q + x ς 1 ς 2 ς 1 ζ + 1 1 ζ + 1 + ς 2 x ς 2 ς 1 | Π ( ς 1 ) | q + x ς 1 ς 2 ς 1 | Π ( ς 2 ) | q 1 q } .
Theorem 13
([9]). Let Π be as in Theorem 10. If | Π | q , q 1 is convex on [ ς 1 , ς 2 ] , and x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) + J x ζ Π ( ς 1 ) | 1 ( ς 2 ς 1 ) ζ + 1 ( ζ p + 1 ) 1 p ( ς 2 x ) ζ + 1 | Π ς 2 + x 2 | + ( x ς 1 ) ζ + 1 | Π ς 1 + x 2 | .
Now we give some weighted fractional Ostrowski-type integral inequalities.
Theorem 14
([10]). Let Π : [ ς 1 , ς 2 ] R be a function which is differentiable on ( ς 1 , ς 2 ) with ς 1 < ς 2 and Π L [ ς 1 , ς 2 ] . If g : [ ς 1 , ς 2 ] R is continuous and | Π | is convex on [ ς 1 , ς 2 ] , then
| J ς 1 + ζ g ( x ) + J ς 2 ζ g ( x ) Π ( x ) J ς 1 + ζ ( Π g ) ( x ) + J ς 2 ζ ( Π g ) ( x ) | g ( ς 2 ς 1 ) Γ ( ζ + 1 ) [ ( x ς 1 ) ζ + 1 ς 2 ς 1 + x 2 ( ζ + 2 ) ( ς 2 x ) + ( ζ + 1 ) ( x ς 1 ) ( ζ + 1 ) ( ζ + 2 ) + ( ς 2 x ) ζ + 2 1 2 1 ( ζ + 1 ) ( ζ + 2 ) ] | Π ( ς 1 ) | + [ ( ς 2 x ) ζ + 1 ς 2 + x 2 ς 1 ( ζ + 1 ) ( ς 2 x ) + ( ζ + 2 ) ( x ς 1 ) ( ζ + 1 ) ( ζ + 2 ) + ( x ς 1 ) ζ + 2 1 2 1 ( ζ + 1 ) ( ζ + 2 ) ] | Π ( ς 2 ) | ,
for all x [ ς 1 , ς 2 ] , where g = sup { | g ( x ) | : x [ ς 1 , ς 2 ] } .
Theorem 15
([10]). Let Π and g be as in Theorem 14. If | Π | q , q > 1 is convex on [ ς 1 , ς 2 ] , then
| J ς 1 + ζ g ( x ) + J ς 2 ζ g ( x ) Π ( x ) J ς 1 + ζ ( Π g ) ( x ) + J ς 2 ζ ( Π g ) ( x ) | g ( ς 2 ς 1 ) 1 q Γ ( ζ + 1 ) 1 1 p ζ + 1 1 p [ ( x ς 1 ) ζ + 1 ( ς 2 ς 1 + x 2 | Π ( ς 1 ) | q + x ς 1 2 | Π ( ς 2 ) | q ) 1 q + ( ς 2 x ) ζ + 1 ς 2 x 2 | Π ( ς 1 ) | q + x + ς 2 2 ς 1 | Π ( ς 2 ) | q 1 q ] ,
for all x [ ς 1 , ς 2 ] , where 1 p + 1 q = 1 .

2.3. Ostrowski-Type Fractional Integral Inequalities for Quasi-Convex Functions

Definition 3
([11]). A real-valued Π is quasi-convex, if
Π λ x + 1 λ y max { Π ( x ) , Π ( y ) } ,
holds for all x , y I and λ [ 0 , 1 ] .
In the following theorems, we explore some weighted Ostrowski-type inequalities in the frame of fractional operator for quasi-convex functions.
Theorem 16
([12]). Let Π : [ ς 1 , ς 2 ] R be a function which is differentiable on ( ς 1 , ς 2 ) where 0 ς 1 < ς 2 and g : [ ς 1 , ς 2 ] R be a continuous function. If | Π | is quasi-convex, then
| J x ζ ( Π g ) ( ς 1 ) + J x + ζ ( Π g ) ( ς 2 ) J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | ( ς 2 x ) ζ + 1 Γ ( ζ + 2 ) max { | Π ( x ) | , | Π ( ς 2 ) | } g [ x , ς 2 ] , + ( x ς 1 ) ζ + 1 Γ ( ζ + 2 ) max { | Π ( x ) | , | Π ( ς 1 ) | } g [ ς 1 , x ] , ,
for all x [ ς 1 , ς 2 ] .
Theorem 17
([12]). Let Π be as in Theorem 16. If | Π | q is quasi-convex, q > 1 and 1 p + 1 q = 1 , then
| J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | ( ς 2 x ) ζ + 1 ( ζ p + 1 ) 1 p Γ ( ζ + 1 ) max { | Π ( x ) | q , | Π ( ς 2 ) | q } 1 q g [ x , ς 2 ] , + ( x ς 1 ) ζ + 1 ( ζ p + 1 ) 1 p Γ ( ζ + 2 ) max { | Π ( x ) | q , | Π ( ς 1 ) | q } 1 q g [ ς 1 , x ] , ,
for all x [ ς 1 , ς 2 ] .
Theorem 18
([12]). Let Π be as in Theorem 16. If | Π | q is quasi-convex, q 1 then
| J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | ( ς 2 x ) ζ + 1 Γ ( ζ + 1 ) max { | Π ( x ) | q , | Π ( ς 2 ) | q } 1 q g [ x , ς 2 ] , + ( x ς 1 ) ζ + 1 Γ ( ζ + 2 ) max { | Π ( x ) | q , | Π ( ς 1 ) | q } 1 q g [ ς 1 , x ] , ,
for all x [ ς 1 , ς 2 ] .
A further result for functions with a bounded first derivative is given in the next theorem.
Theorem 19
([12]). Let the assumptions of this theorem be as stated in Theorem 16. If there exist constants m < M such that < m Π ( x ) M < + for all x [ ς 1 , ς 2 ] , then
| J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | ( M + m ) ( ς 2 x ) ζ + 1 ( x ς 1 ) ζ + 1 2 Γ ( ζ ) 0 1 ( p 1 ( τ ) + p 2 ( τ ) ) d τ ( M m ) ( ς 2 x ) ζ + 1 2 Γ ( ζ + 2 ) g [ x , ς 2 ] , + ( M m ) ( x ς 1 ) ζ + 1 2 Γ ( ζ + 2 ) g [ ς 1 , x ] , ,
where
p 1 ( τ ) = τ 1 ( 1 σ ) ζ 1 g ( σ ς 2 + ( 1 σ ) x ) d σ , p 2 ( τ ) = τ 1 ( 1 σ ) ζ 1 g ( σ ς 1 + ( 1 σ ) x ) d σ .
Definition 4
([13]). A function Π : I R is said to be a strongly quasi-convex function with modulus c 0 , if
Π ( t x + ( 1 t ) y ) max { Π ( x ) , Π ( y ) } c t ( 1 t ) ( y x ) 2 , x , y I , t [ 0 , 1 ] .
The aim of this subsection is to give some Ostrowski-type fractional integral inequalities for strongly quasi-convex functions.
Theorem 20
([14]). Let Π : [ ς 1 , ς 2 ] [ 0 , ) R be a differentiable mapping on ( ς 1 , ς 2 ) such that Π L [ ς 1 , ς 2 ] . If | Π | is a strongly quasi-convex function with modulus c 0 , on [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( s ) d s | ( ς 2 x ) 2 2 ( ς 2 ς 1 ) max { | Π ( x ) | , | Π ( ς 2 ) | } c ( ς 2 x ) 3 3 ( ς 2 ς 1 ) 2 ( ς 2 x ) 4 4 ( ς 2 ς 1 ) 3 ( x ς 2 ) 2 + ( x ς 1 ) 2 2 ( ς 2 ς 1 ) max { | Π ( x ) | , | Π ( ς 1 ) | } c ( x ς 1 ) 2 1 12 ( ς 2 x ) 2 2 ( ς 2 ς 1 ) + 2 ( ς 2 x ) 3 3 ( ς 2 ς 1 ) 2 ( ς 2 x ) 4 4 ( ς 2 ς 1 ) 3 ,
for each x [ ς 1 , ς 2 ] .
Theorem 21
([14]). Let Π be as in Theorem 20. If | Π | q is a strongly quasi-convex function with modulus c 0 , on [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( s ) d s | ( ς 2 x ) p + 1 ( ς 2 ς 1 ) ( p + 1 ) max { | Π ( x ) | q , | Π ( ς 2 ) | q } c ( ς 2 x ) 2 2 ( ς 2 ς 1 ) 2 ( ς 2 x ) 3 3 ( ς 2 ς 1 ) 3 ( x ς 2 ) 2 1 q + ( x ς 1 ) p + 1 ( ς 2 ς 1 ) ( p + 1 ) max { | Π ( x ) | q , | Π ( ς 1 ) | q } c 1 6 ( ς 2 x ) 2 2 ( ς 2 ς 1 ) 2 + ( ς 2 x ) 3 3 ( ς 2 ς 1 ) 3 ( x ς 1 ) 2 1 q ,
for each x [ ς 1 , ς 2 ] , q > 1 and 1 p + 1 q = 1 .
Theorem 22
([14]). Let Π be as in Theorem 20. If | Π | q is a strongly quasi-convex function with modulus c 0 , on [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( s ) d s | ( ς 2 x ) 2 2 ( ς 2 ς 1 ) max { | Π ( x ) | q , | Π ( ς 2 ) | q } c 2 ( ς 2 x ) 3 ( ς 2 ς 1 ) ( ς 2 x ) 2 2 ( ς 2 ς 1 ) 2 ( x ς 2 ) 2 1 q + ( x ς 1 ) 2 2 ( ς 2 ς 1 ) ( max { | Π ( x ) | q , | Π ( ς 1 ) | q } c ( ( ς 2 x ) 2 6 ( x ς 1 ) 2 ( ς 2 x ) 2 ( x ς 1 ) 2 + 4 ( ς 2 x ) 3 3 ( ς 2 ς 1 ) ( x ς 1 ) 2 ( ς 2 x ) 4 2 ( ς 2 ς 1 ) 2 ( x ς 1 ) 2 ) ( x ς 1 ) 2 ) 1 q ,
for each x [ ς 1 , ς 2 ] .
In the next, we present fractional weighted Ostrowski-type fractional integral inequalities via a strongly quasi-convex function.
Theorem 23
([14]). Let Π be as in Theorem 20 and g : [ ς 1 , ς 2 ] R be a continuous function. If | Π | is a strongly quasi-convex function with modulus c 0 , on [ ς 1 , ς 2 ] , then
| J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) Π ( ς 1 ) | ( ς 2 x ) ζ + 1 Γ ( ζ + 1 ) max { | Π ( x ) | , | Π ( ς 2 ) | } g [ x , ς 2 ] , ( ς 2 x ) ζ + 1 Γ ( ζ + 3 ) ( ς 2 x ) ζ + 1 Γ ( ζ + 4 ) c ( x ς 2 ) 2 g [ x , ς 2 ] , + ( x ς 1 ) ζ + 1 Γ ( ζ + 2 ) max { | Π ( x ) | , | Π ( ς 1 ) | } g [ ς 1 , x ] , ( ( x ς 1 ) ζ + 1 Γ ( ζ + 3 ) ( x ς 1 ) ζ + 1 Γ ( ζ + 4 ) c ( x ς 1 ) 2 ) g [ x , ς 2 ] , ,
for each x [ ς 1 , ς 2 ] .
Theorem 24
([14]). Let Π be as in Theorem 20 and g : [ ς 1 , ς 2 ] R be a continuous function. If | Π | q is a strongly quasi-convex function with modulus c 0 , q > 1 and 1 p + 1 q = 1 , then
| J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) J x ζ g Π ( ς 1 ) + J x + ζ g Π ( ς 2 ) Π ( ς 1 ) | ( ς 2 x ) ζ + 1 ( ζ p + 1 ) 1 p Γ ( ζ + 1 ) g [ x , ς 2 ] , max { | Π ( x ) | q , | Π ( ς 2 ) | q } c 6 ( x ς 2 ) 2 1 q + ( x ς 1 ) ζ + 1 ( ζ p + 1 ) 1 p Γ ( ζ + 1 ) g [ ς 1 , x ] , max { | Π ( x ) | q , | Π ( ς 1 ) | q } c 6 ( x ς 1 ) 2 1 q ,
for each x [ ς 1 , ς 2 ] .

2.4. Ostrowski-Type Fractional Integral Inequalities for ( ζ , m ) -Convex Functions

Definition 5
([15]). The function Π : [ 0 , b ] R , b > 0 is said to be ( ζ , m ) -convex, if
Π ( t x + ( 1 t ) y ) t ζ Π ( x ) + m ( 1 t ζ ) Π ( y ) ,
for all x , y [ 0 , b ] , ( ζ , m ) [ 0 , 1 ] 2 and t [ 0 , 1 ] .
Ostrowski-type fractional integral inequalities pertaining to Riemann–Liouville fractional integral for ( ζ , m ) -convex functions are presented in the following theorems.
Theorem 25
([16]). Let I be an open real interval such that [ 0 , ) I and Π : I R be a differentiable mapping on I such that Π L [ m ς 1 , m ς 2 ] , where m ς 1 , m ς 2 I with ς 1 < ς 2 , m ( 0 , 1 ] . If | Π | is ( ζ , m ) -convex on [ m ς 1 , m ς 2 ] for ( ζ , m ) [ 0 , 1 ] 2 and | Π ( x ) | M , then
| ( x m ς 1 ) ζ + ( m ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 I x ζ Π ( m ς 1 ) + I x ζ Π ( m ς 1 ) M ( x m ς 1 ) ζ + 1 + ( m ς 2 x ) ζ + 1 ς 2 ς 1 1 + m ζ 1 + 2 ζ ,
for all x [ m ς 1 , m ς 2 ] .
Theorem 26
([16]). Let Π be as in Theorem 25. If | Π | q , q > 1 is ( ζ , m ) -convex on [ m ς 1 , m ς 2 ] for ( ζ , m ) [ 0 , 1 ] 2 and | Π ( x ) | M , then
| ( x m ς 1 ) ζ + ( m ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 I x ζ Π ( m ς 1 ) + I x ζ Π ( m ς 1 ) M 1 ζ p + 1 1 p ( x m ς 1 ) ζ + 1 + ( m ς 2 x ) ζ + 1 ς 2 ς 1 1 + m ζ 1 + ζ 1 q ,
with 1 p + 1 q = 1 and x [ m ς 1 , m ς 2 ] .
Theorem 27
([16]). Let Π be as in Theorem 25. If | Π | q , q 1 is ( ζ , m ) -convex on [ m ς 1 , m ς 2 ] for ( ζ , m ) [ 0 , 1 ] 2 and | Π ( x ) | M , then
| ( x m ς 1 ) ζ + ( m ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 I x ζ Π ( m ς 1 ) + I x ζ Π ( m ς 1 ) M ( x m ς 1 ) ζ + 1 + ( m ς 2 x ) ζ + 1 ( ς 2 ς 1 ) ( ζ + 1 ) 1 + ζ ( m + 1 ) 2 ζ + 1 1 q ,
for all x [ m ς 1 , m ς 2 ] .

2.5. Ostrowski-Type Fractional Integral Inequalities for s-Convex Functions

Definition 6
([17]). A function Π : [ 0 , ) R is said to be s-convex in the second sense, if
Π λ ς 1 + 1 λ ς 2 λ s Π ς 1 + 1 λ s Π ς 2 ,
holds for all ς 1 , ς 2 [ 0 , ) , λ [ 0 , 1 ] and for some fixed s ( 0 , 1 ] .
Ostrowski-type inequalities pertaining to Riemann–Liouville fractional integral for s-convex functions are presented.
Theorem 28
([18]). Let Π : [ ς 1 , ς 2 ] [ 0 , ) R be a function which is differentiable on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . Suppose | Π | is s-convex in the second sense on [ ς 1 , ς 2 ] for s ( 0 , 1 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] . Then, for all x [ ς 1 , ς 2 ] ,
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ς 2 ς 1 1 + Γ ( ζ + 1 ) Γ ( s + 1 ) Γ ( ζ + s + 1 ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ζ + s + 1 .
Theorem 29
([18]). Let Π be as in Theorem 28. If | Π | q , q > 1 is s-convex in the second sense on [ ς 1 , ς 2 ] for s ( 0 , 1 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( 1 + ζ p ) 1 p 2 s + 1 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 , f o r   a l l   x [ ς 1 , ς 2 ] ,
where ζ > 0 and 1 p + 1 q = 1 .
Theorem 30
([18]). Let Π be as in Theorem 28. If | Π | q , q 1 is s-convex in the second sense on [ ς 1 , ς 2 ] for s ( 0 , 1 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 1 + ζ 1 1 q 1 ζ + s + 1 1 q 1 + Γ ( ζ + 1 ) Γ ( s + 1 ) Γ ( ζ + s + 1 ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 ,
for all x [ ς 1 , ς 2 ] , with ζ > 0 .
Theorem 31
([19]). Let Π be as in Theorem 28. If | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | 1 ζ + s + 1 ς 2 x ς 2 ς 1 ζ + s + 1 + B ς 2 x ς 2 ς 1 ; ζ + 1 , s + 1 | Π ( ς 1 ) | + 1 ζ + s + 1 x ς 1 ς 2 ς 1 ζ + s + 1 + B ς 2 x ς 2 ς 1 ; ζ + 1 , s + 1 | Π ( ς 2 ) | ,
for all x ( ς 1 , ς 2 ) .
Theorem 32
([19]). Let Π be as in Theorem 28. If | Π | q , q > 1 , 1 p + 1 q = 1 is s-convex in the second sense on [ ς 1 , ς 2 ] for s ( 0 , 1 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | 1 ζ p + 1 1 p ς 2 x ς 2 ς 1 ζ + 1 p 1 s + 1 ς 2 x ς 2 ς 1 s + 1 | Π ( ς 1 ) | q + 1 x ς 1 ς 2 ς 1 s + 1 | Π ( ς 2 ) | q 1 q + 1 ζ p + 1 1 p x ς 1 ς 2 ς 1 ζ + 1 p 1 s + 1 1 ς 2 x ς 2 ς 1 s + 1 | Π ( ς 1 ) | q + x ς 1 ς 2 ς 1 s + 1 | Π ( ς 2 ) | q 1 q ,
for all x ( ς 1 , ς 2 ) .
Theorem 33
([19]). Let Π be as in Theorem 28. If | Π | q , q > 1 , 1 p + 1 q = 1 is s-convex in the second sense on [ ς 1 , ς 2 ] for s ( 0 , 1 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ( ς 2 ς 1 ) ζ + 1 Π ( x ) Γ ( ζ + 1 ) ( ς 2 ς 1 ) ζ + 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | 1 ζ p + 1 1 p ς 2 x ς 2 ς 1 ζ + 1 | Π ( x ) | q + | Π ( ς 2 ) | q 2 1 q + x ς 1 ς 2 ς 1 ζ + 1 | Π ( x ) | q + | Π ( ς 1 ) | q 2 1 q ,
for all x ( ς 1 , ς 2 ) .

2.6. Ostrowski-Type Fractional Integral Inequalities for ( s , r ) -Convex Functions

Definition 7
([20]). A function Π : I [ 0 , ) [ 0 , ) is said to be ( s , r ) -convex in mixed kind, if
Π λ x + 1 λ y λ r s Π ( x ) + 1 λ r s Π ( y ) ,
holds for all x , y I , λ [ 0 , 1 ] and ( s , r ) [ 0 , 1 ] 2 .
Now, we state the generalization of the classical Ostrowski inequality via fractional integrals, which is obtained for ( s , r ) -convex function in mixed kind.
Theorem 34
([20]). Let Π : [ ς 1 , ς 2 ] R be a function which is differentiable on ( ς 1 , ς 2 ) with ς 1 < ς 2 and Π L [ ς 1 , ς 2 ] . If | Π | is ( s , r ) -convex on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 ζ + r s + 1 + B ( ζ + 1 r , s + 1 ) r ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ζ + s + 1 ,
for all x ( ς 1 , ς 2 ) .
Theorem 35
([20]). Let Π be as in Theorem 34. If | Π | q is ( s , r ) -convex on [ ς 1 , ς 2 ] , q 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( ζ + 1 ) 1 1 q 1 ζ + r s + 1 + B ( ζ + 1 r , s + 1 ) r 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ζ + s + 1 ,
for all x ( ς 1 , ς 2 ) .
Theorem 36
([20]). Let Π be as in Theorem 34. If | Π | q be ( s , r ) -convex on [ ς 1 , ς 2 ] , q > 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( ζ p + 1 ) 1 p 1 r s + 1 + B ( 1 r , s + 1 ) r 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ζ + s + 1 ,
for all x ( ς 1 , ς 2 ) and 1 p + 1 q = 1 .

2.7. Ostrowski-Type Fractional Integral Inequalities for Harmonically-Convex Functions

Definition 8
([21]). Let I R { 0 } be a real interval. A function Π : I R is harmonically convex, if
Π x y t x + ( 1 t ) y t Π ( y ) + ( 1 t ) Π ( x )
for all x , y I and t [ 0 , 1 ] .
Some new Ostrowski’s-type fractional integral inequalities for functions whose first derivatives are harmonically convex, via Riemann–Liouville fractional integrals are given in the next theorems.
Theorem 37
([22]). Let Π : [ ς 1 , ς 2 ] ( 0 , ) R be a differentiable mapping on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | is harmonically convex on [ ς 1 , ς 2 ] , then, for all x [ ς 1 , ς 2 ] ,
| Γ ( ζ + 1 ) 2 ς 1 ς 2 ς 2 ς 1 ζ J 1 ς 1 ζ ( Π h ) 1 ς 2 + J 1 ς 2 + ζ ( Π h ) 1 ς 1 Π ( x ) | ς 1 ς 2 ( ς 2 ς 1 ) 2 μ 1 ( ς 1 , ς 2 , ζ ) | Π ( ς 1 ) | + μ 2 ( ς 1 , ς 2 , ζ ) | Π ( ς 2 ) | ,
where
μ 1 ( ς 1 , ς 2 , ζ ) = 1 ( ς 2 ς 1 ) 2 ς 1 ς 2 1 + ln ς 2 ς 1 + 2 F 1 2 , 2 ; ζ + 3 ; 1 2 1 ς 2 ς 1 4 ς 1 2 ( ζ + 1 ) ( ζ + 2 ) + 2 F 1 2 , 1 ; ζ + 2 ; 1 2 1 ς 1 ς 2 2 ς 2 2 ( ζ + 1 ) 2 F 1 2 , 2 ; ζ + 3 ; 1 2 1 ς 1 ς 2 4 ς 2 2 ( ζ + 1 ) ( ζ + 2 ) , μ 2 ( ς 1 , ς 2 , ζ ) = 1 ( ς 2 ς 1 ) 2 ς 2 ς 1 1 + ln ς 1 ς 2 + 2 F 1 2 , 1 ; ζ + 2 ; 1 2 1 ς 2 ς 1 2 ς 1 2 ( ζ + 1 ) + 2 F 1 2 , 2 ; ζ + 3 ; 1 2 1 ς 1 ς 2 4 ς 2 2 ( ζ + 1 ) ( ζ + 2 ) 2 F 1 2 , 2 ; ζ + 3 ; 1 2 1 ς 2 ς 1 4 ς 2 2 ( ζ + 1 ) ( ζ + 2 ) ,
with 2 F 1 ( . , . ; . ; . ) the hypergeometric function and h ( x ) = 1 x , x 1 ς 2 , 1 ς 1 .
Theorem 38
([22]). Let Π : [ ς 1 , ς 2 ] ( 0 , ) R be a differentiable mapping on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | q is harmonically convex on [ ς 1 , ς 2 ] , where q > 1 and 1 p + 1 q = 1 , then, for all x [ ς 1 , ς 2 ] ,
| Γ ( ζ + 1 ) 2 ς 1 ς 2 ς 2 ς 1 ζ J 1 ς 1 ζ ( Π h ) 1 ς 2 + J 1 ς 2 + ζ ( Π h ) 1 ς 1 Π ( x ) | ς 1 ς 2 ( ς 2 ς 1 ) 2 [ 1 ( ς 2 ς 1 ) 1 p ( 2 p 1 ) 1 p 1 ς 1 2 p 1 1 ς 2 2 p 1 1 p | Π ( ς 1 ) | q + | Π ( ς 2 ) | q 2 1 q + 1 2 1 p ς 1 2 2 F 1 2 p , 1 ; ζ p + 2 ; 1 2 1 ς 2 ς 1 ζ p + 1 1 p | Π ( ς 1 ) | q + 3 | Π ( ς 2 ) | q 4 1 q + 1 2 1 p ς 2 2 2 F 1 2 p , 1 ; ζ p + 2 ; 1 2 1 ς 1 ς 2 ζ p + 1 1 p 3 | Π ( ς 1 ) | q + | Π ( ς 2 ) | q 4 1 q ] .

2.8. Ostrowski-Type Fractional Integral Inequalities for h-Convex Functions

Definition 9
([23]). Suppose h is a non-negative and real-valued function. Then Π : I R is an h-convex, if Π is non-negative and for all x , y I , λ ( 0 , 1 ) we have
Π ( λ x + ( 1 λ ) y ) h ( λ ) Π ( x ) + h ( 1 λ ) Π ( y ) .
Some Ostrowski-type inequalities via Riemann–Liouville fractional integrals for h-convex are given in the next theorems.
Theorem 39
([24]). Let Π : [ ς 1 , ς 2 ] R be a function which is differentiable on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 . If | Π | is h-convex on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| Π ( x ) Γ ( ζ + 1 ) 1 2 ( x ς 1 ) ζ J x ζ Π ( ς 1 ) + 1 2 ( ς 2 x ) ζ J x + ζ Π ( ς 2 ) | M ( ς 2 ς 1 ) 2 0 1 [ h ( 1 t ) + h ( t ) ] t ζ d t ,
for each x [ ς 1 , ς 2 ] .
Theorem 40
([24]). Let Π be as in the Theorem 39. If | Π | q is h-convex on [ ς 1 , ς 2 ] , p , q > 1 , 1 p + 1 q = 1 , then
| Π ( x ) Γ ( ζ + 1 ) 1 2 ( x ς 1 ) ζ J x ζ Π ( ς 1 ) + 1 2 ( ς 2 x ) ζ J x + ζ Π ( ς 2 ) | M ( ς 2 ς 1 ) 2 ( ζ p + 1 ) 1 p 2 0 1 h ( t ) d t 1 q ,
for each x [ ς 1 , ς 2 ] .
Theorem 41
([24]). Let Π be as in the Theorem 39. If | Π | q is h-convex on [ ς 1 , ς 2 ] , q 1 , then
| Π ( x ) Γ ( ζ + 1 ) 1 2 ( x ς 1 ) ζ J x ζ Π ( ς 1 ) + 1 2 ( ς 2 x ) ζ J x + ζ Π ( ς 2 ) | M ( ς 2 ς 1 ) 2 1 ζ + 1 1 1 q 0 1 t ζ [ h ( t ) + h ( 1 t ) ] d t 1 q ,
for each x [ ς 1 , ς 2 ] .
Ostrowski-type fractional integral inequalities for super-multiplicative functions pertaining to Riemann–Liouville fractional integrals are given now.
Definition 10
([25]). We say that h : J R is a super-multiplicative function, if for all x , y J , one has
h ( x , y ) h ( x ) h ( y ) .
Theorem 42
([26]). Let h : J R R ( [ 0 , 1 ] J ) be a super-multiplicative and non-negative function, h ( t ) t for 0 t 1 , Π : [ ς 1 , ς 2 ] ( 0 , ) R be a differentiable function on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | is a h-convex function on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 0 1 [ t ζ h ( t ) + t ζ h ( 1 t ) ] d t M ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 0 1 [ h ( t ζ + 1 ) + h ( t ζ ( 1 t ) ] d t .
Theorem 43
([26]). Let Π be as in Theorem 42. If | Π | q is a h-convex function on [ ς 1 , ς 2 ] , p , q > 1 1 p + 1 q = 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ( 1 + p ζ ) 1 p ( ς 2 ς 1 ) 0 1 [ h ( t ) + h ( 1 t ) ] d t 1 q M ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ( 1 + p ζ ) 1 p ( ς 2 ς 1 ) h 1 q ( 1 ) .
Theorem 44
([26]). Let Π be as in Theorem 42. If | Π | q , q 1 is a h-convex function on [ ς 1 , ς 2 ] , and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( 1 + ζ ) 1 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 0 1 [ t ζ h ( t ) + t ζ h ( 1 t ) ] d t 1 q M ( 1 + ζ ) 1 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 0 1 [ h ( t ζ + 1 ) + h ( t ζ ( 1 t ) ) ] d t 1 q .

2.9. Ostrowski-Type Fractional Integral Inequalities for Godunova-Levin Functions

Definition 11
([27]). A function Π : [ ς 1 , ς 2 ] R is a Godunova–Levin function if
Π ( t x + ( 1 t ) y ) Π ( x ) t + Π ( y ) 1 t ,
for all x , y [ ς 1 , ς 2 ] and t [ 0 , 1 ] .
Definition 12
([28]). A function Π : [ ς 1 , ς 2 ] R is an s-Godunova-Levin function of the first kind, where s ( 0 , 1 ] , if
Π ( t x + ( 1 t ) y ) Π ( x ) t s + Π ( y ) 1 t s ,
for all x , y [ ς 1 , ς 2 ] and t ( 0 , 1 ) .
Definition 13
([28]). A function Π : [ ς 1 , ς 2 ] R is said to be an s-Godunova-Levin function of the second kind, where s ( 0 , 1 ] , if
Π ( t x + ( 1 t ) y ) Π ( x ) t s + Π ( y ) ( 1 t ) s ,
for all x , y [ ς 1 , ς 2 ] and t ( 0 , 1 ) .
In this subsection, we show some Ostrowski-type inequalities pertaining to Riemann–Liouville fractional integrals for s-Godunova-Levin functions.
Theorem 45
([29]). Suppose Π : [ ς 1 , ς 2 ] R is a differentiable function on ( ς 1 , ς 2 ) with ς 1 < ς 2 and Π L [ ς 1 , ς 2 ] . If | Π | is an s-Godunova-Levin function of the second kind on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 1 + ζ s + Γ ( 1 s ) Γ ( ζ + 1 ) Γ ( 2 + ζ s ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 ,
for all x ( ς 1 , ς 2 ) .
Theorem 46
([29]). Let Π be as in Theorem 45. If | Π | q is an s-Godunova-Levin function of the second kind on [ ς 1 , ς 2 ] , p , q > 1 , 1 p + 1 q = 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 1 s 1 q 1 1 + p ζ 1 p ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 ,
for all x ( ς 1 , ς 2 ) .
Now, we present some new family of s-Godunova-Levin functions, which are called ( s , m ) -Godunova-Levin functions of the second kind. Next, we present some new Ostrowski-type integral inequalities for ( s , m ) -Godunova-Levin functions via fractional integrals.
Definition 14
([30]). A function Π : [ ς 1 , ς 2 ] R is said to be an ( s , m ) -Godunova-Levin function of the second kind, where s [ 0 , 1 ] , m ( 0 , 1 ] , if
Π ( t x + ( 1 t ) y ) Π ( x ) t s + m Π ( y ) ( 1 t ) s ,
for all x , y [ ς 1 , ς 2 ] and t ( 0 , 1 ) .
Theorem 47
([30]). Suppose Π : [ ς 1 , ς 2 ] R is a differentiable function on open interval ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | is an ( s , m ) -Godunova-Levin function of the second kind on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then, for all x [ ς 1 , ς 2 ] ,
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | min { ϑ 1 ( ς 1 , ς 2 ; m , ζ ; x ) , ϑ 2 ( ς 1 , ς 2 ; m , ζ ; x ) } ,
where
ϑ 1 ( ς 1 , ς 2 ; m , ζ ; x ) = M 1 + ζ s ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 + m Γ ( 1 s ) Γ ( ζ + 1 ) Γ ( 2 + ζ s ) ( x ς 1 ) ζ + 1 | Π ς 2 m | + ( ς 2 x ) ζ + 1 | Π ς 1 m | ς 2 ς 1 , ϑ 2 ( ς 1 , ς 2 ; m , ζ ; x ) = m 1 + ζ s | Π x m | + M Γ ( 1 s ) Γ ( ζ + 1 ) Γ ( 2 + ζ s ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 48
([30]). Let Π be as in Theorem 47. Then, for all x [ ς 1 , ς 2 ] ,
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | min { φ 1 ( ς 1 , ς 2 ; m , ζ ; x ) , φ 2 ( ς 1 , ς 2 ; m , ζ ; x ) } ,
where
φ 1 ( ς 1 , ς 2 ; m , ζ ; x ) = 1 p ζ + 1 1 p [ M q 1 s + m 1 s | Π ς 1 m | q 1 q ( x ς 1 ) ζ + 1 ς 2 ς 1 + M q 1 s + m 1 s | Π ς 2 m | q 1 q ( ς 2 x ) ζ + 1 ς 2 ς 1 ] , φ 2 ( ς 1 , ς 2 ; m , ζ ; x ) = 1 p ζ + 1 1 p m 1 s | Π x m | q + M q 1 s 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 49
([30]). Let the assumptions of this theorem be stated in Theorem 47. Then, for all x [ ς 1 , ς 2 ] ,
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | min { ρ 1 ( ς 1 , ς 2 ; m , ζ ; x ) , ρ 2 ( ς 1 , ς 2 ; m , ζ ; x ) } ,
where
ρ 1 ( ς 1 , ς 2 ; m , ζ ; x ) = 1 ζ + 1 1 1 q [ ( x ς 1 ) ζ + 1 ς 2 ς 1 M q 1 + ζ s + m Γ ( 1 s ) Γ ( ζ + 1 ) Γ ( 2 + ζ s ) | Π ς 1 m | q 1 q + ( ς 2 x ) ζ + 1 ς 2 ς 1 M q 1 + ζ s + m Γ ( 1 s ) Γ ( ζ + 1 ) Γ ( 2 + ζ s ) | Π ς 2 m | q 1 q ] , ρ 2 ( ς 1 , ς 2 ; m , ζ ; x ) = 1 ζ + 1 1 1 q [ ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 × m 1 + ζ s | Π x m | q + M q Γ ( 1 s ) Γ ( ζ + 1 ) Γ ( 2 + ζ s ) 1 q ] .

2.10. Ostrowski-Type Fractional Integral Inequalities for M T -Convex Function

Definition 15
([31]). A real-valued and non-negative function Π is M T -convex function, if
Π ( t x + ( 1 t ) y ) t 2 1 t Π ( x ) + 1 t 2 t Π ( y ) ,
for all x , y I and t ( 0 , 1 ) .
In this subsection, we give some Ostrowski-type fractional integral inequalities for M T -convex functions via Riemann–Liouville fractional integrals.
Theorem 50
([32]). Suppose Π : [ ς 1 , ς 2 ] ( 0 , ) R is a mapping which is differentiable on ( ς 1 , ς 2 ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | is MT-convex function on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M Γ ζ + 1 2 Γ 1 2 2 Γ ( ζ + 1 ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 51
([32]). Let Π be as in Theorem 50. If | Π | q is MT-convex function on [ ς 1 , ς 2 ] , q > 1 , 1 p + 1 q = 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( 1 + p ζ ) 1 p π 2 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 52
([32]). Let Π be as in Theorem 50. If | Π | q , q 1 is MT-convex function on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( 1 + p ζ ) 1 p Γ ζ + 1 2 Γ 1 2 2 Γ ( ζ + 1 ) 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 53
([33]). Let the assumptions of this theorem be stated in Theorem 50. Then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | 2 M B ζ + 1 2 , 1 2 ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 54
([33]). Let Π be as in Theorem 50. If | Π | q is MT-convex function on [ ς 1 , ς 2 ] , q > 1 , 1 p + 1 q = 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M ( 1 + p ζ ) 1 p π 4 1 q ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 ς 2 ς 1 .
Theorem 55
([34]). Let the assumptions of this theorem be stated in Theorem 50. Then for ζ > 0 , λ [ 0 , 1 ] and x [ ς 1 , ς 2 ] we have:
| ( 1 λ ) ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) + λ ( x ς 1 ) ζ Π ( ς 1 ) + ( ς 2 x ) ζ Π ( ς 2 ) ς 2 ς 1 Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 2 ( ς 2 ς 1 ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 A ( ζ , λ ) ,
where
A ( ζ , λ ) = 2 λ ( B λ 1 ζ ; 3 2 , 1 2 + B λ 1 ζ ; 1 2 , 3 2 + B ζ + 1 2 , 1 2 λ π 2 B λ 1 ζ ; ζ + 3 2 , 1 2 + B λ 1 ζ ; ζ + 1 2 , 3 2 ,
and B ( a , ; x , y ) = 0 a t x 1 ( 1 t ) y 1 d t , 0 < a 1 , x , y > 0 the incomplete Beta function.
Theorem 56
([34]). Let Π be as in Theorem 50. If | Π | q is MT-convex function on [ ς 1 , ς 2 ] , q > 1 , 1 p + 1 q = 1 and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( 1 λ ) ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) + λ ( x ς 1 ) ζ Π ( ς 1 ) + ( ς 2 x ) ζ Π ( ς 2 ) ς 2 ς 1 Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 2 ( ς 2 ς 1 ) ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 π 2 1 q B ( ζ , λ ) 1 p ,
where
B ( ζ , λ ) = 2 ζ 0 λ ( λ s ) p s 1 ζ 1 d s 1 ζ 0 1 ( λ s ) p s 1 ζ 1 d s .
Theorem 57
([34]). Let Π be as in Theorem 50. If | Π | q , q 1 is MT-convex function on [ ς 1 , ς 2 ] and | Π ( x ) | M , x [ ς 1 , ς 2 ] , then for ζ > 0 and x [ ς 1 , ς 2 ] we have:
| ( 1 λ ) ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) + λ ( x ς 1 ) ζ Π ( ς 1 ) + ( ς 2 x ) ζ Π ( ς 2 ) ς 2 ς 1 Γ ( ζ + 1 ) ς 2 ς 1 J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M 1 ς 2 ς 1 ( x ς 1 ) ζ + 1 + ( ς 2 x ) ζ + 1 A ( ζ , λ ) 2 1 q 2 ζ λ 1 + 1 ζ + 1 ζ + 1 λ 1 1 q ,
where A ( ζ , λ ) is given in Theorem 55.

2.11. Ostrowski-Type Fractional Integral Inequalities for P-Convex, m-Convex and ( s , m ) -Convex Functions

In this subsection, we show results on Ostrowski-type fractional integral inequalities for twice differentiable functions and different kinds of convexity.
Definition 16
([35]). The function Π : I R R is said to be P-convex, if is nonnegative and
Π ( t x + ( 1 t ) y ) Π ( x ) + Π ( y ) ,
x , y I and t [ 0 , 1 ] .
Definition 17
([36]). A real-valued function Π is m-convex, if
Π ( t x + m ( 1 t ) y ) t Π ( x ) + m ( 1 t ) Π ( y ) ,
x , y ( 0 , b ] t [ 0 , 1 ] and m ( 0 , 1 ] .
Definition 18
([15]). A real-valued function Π is ( s , m ) -convex, if
Π ( t x + m ( 1 t ) y ) t s Π ( x ) + m ( 1 t s ) Π ( y ) ,
x , y ( 0 , b ] t [ 0 , 1 ] and ( s , m ) ( 0 , 1 ] 2 .
Theorem 58
([37]). Let Π : I R be a twice differentiable function on I such that | Π | L [ ς 1 , ς 2 ] , where ς 1 , ς 2 I , with ς 1 < ς 2 . If | Π | is a convex function on [ ς 1 , ς 2 ] , and Π is bounded, i.e., Π = sup x [ ς 1 , ς 2 ] | Π ( x ) | < , for any x [ ς 1 , ς 2 ] , then
| ( ζ + 1 ) ( ς 2 x ) ζ ( x ς 1 ) ζ ( ς 2 ς 1 ) Π ( x ) Γ ( ζ + 2 ) [ ( ς 2 x ) ζ + 1 J x ζ Π ( ς 1 ) + ( x ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) | ( x ς 1 ) ζ + 1 ( ς 2 x ) ζ + 1 ( ς 2 ς 1 ) ζ + 2 Π .
Theorem 59
([37]). Let Π be as in Theorem 58. If | Π | is a P-convex function on [ ς 1 , ς 2 ] , and Π is bounded, i.e., Π = sup x [ ς 1 , ς 2 ] | Π ( x ) | < , for any x [ ς 1 , ς 2 ] , then
| ( ζ + 1 ) ( ς 2 x ) ζ ( x ς 1 ) ζ ( ς 2 ς 1 ) Π ( x ) Γ ( ζ + 2 ) [ ( ς 2 x ) ζ + 1 J x ζ Π ( ς 1 ) + ( x ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) | 2 ( x ς 1 ) ζ + 1 ( ς 2 x ) ζ + 1 ( ς 2 ς 1 ) ζ + 2 Π .
Theorem 60
([37]). Let Π be as in Theorem 58. If | Π | is s-convex on [ ς 1 , ς 2 ] , and Π is bounded, i.e., Π = sup x [ ς 1 , ς 2 ] | Π ( x ) | < , for any x [ ς 1 , ς 2 ] , then
| ( ζ + 1 ) ( ς 2 x ) ζ ( x ς 1 ) ζ ( ς 2 ς 1 ) Π ( x ) Γ ( ζ + 2 ) [ ( ς 2 x ) ζ + 1 J x ζ Π ( ς 1 ) + ( x ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) | ( ς 2 ς 1 ) ( x ς 1 ) ζ + 1 ( ς 2 x ) ζ + 1 1 ζ + s + 2 + B ( ζ + 2 , s + 1 ) Π .
Theorem 61
([37]). Let Π be as in Theorem 58. If | Π | is h-convex on [ ς 1 , ς 2 ] , and Π is bounded, i.e., Π = sup x [ ς 1 , ς 2 ] | Π ( x ) | < , for any x [ ς 1 , ς 2 ] , then
| ( ζ + 1 ) ( ς 2 x ) ζ ( x ς 1 ) ζ ( ς 2 ς 1 ) Π ( x ) Γ ( ζ + 2 ) [ ( ς 2 x ) ζ + 1 J x ζ Π ( ς 1 ) + ( x ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) | Π ( ς 2 ς 1 ) ( x ς 1 ) ζ + 1 ( ς 2 x ) ζ + 1 0 1 ( t ζ + 1 + ( 1 t ) ζ + 1 ) h ( t ) d t .
Theorem 62
([37]). Let Π be as in Theorem 58. If | Π | is m-convex on [ ς 1 , ς 2 ] , and Π is bounded, i.e., Π = sup x [ ς 1 , ς 2 ] | Π ( x ) | < , for any x [ ς 1 , ς 2 ] , then
| ( ζ + 1 ) ( ς 2 x ) ζ ( x ς 1 ) ζ ( ς 2 ς 1 ) Π ( x ) Γ ( ζ + 2 ) [ ( ς 2 x ) ζ + 1 J x ζ Π ( ς 1 ) + ( x ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) | Π ( 1 m ) ( ς 2 ς 1 ) ( x ς 1 ) ζ + 1 ( ς 2 x ) ζ + 1 { ( x ς 1 ) [ 1 ζ + 3 x ς 1 x m ς 1 + 1 ζ + 2 ( 1 m ) ς 1 x m ς 1 + m 1 m ] + ( ς 2 x ) 1 ζ + 3 ς 2 x ς 2 m x + 1 ζ + 2 1 1 m } .
Theorem 63
([37]). Let Π be as in Theorem 58. If | Π | is ( s , m ) -convex on [ ς 1 , ς 2 ] , ( s , m ) ( 0 , 1 ] 2 and Π is bounded, i.e., Π = sup x [ ς 1 , ς 2 ] | Π ( x ) | < , for any x [ ς 1 , ς 2 ] , then
| ( ζ + 1 ) ( ς 2 x ) ζ ( x ς 1 ) ζ ( ς 2 ς 1 ) Π ( x ) Γ ( ζ + 2 ) [ ( ς 2 x ) ζ + 1 J x ζ Π ( ς 1 ) + ( x ς 1 ) ζ + 1 J x + ζ Π ( ς 2 ) | Π ( 1 m ) [ m ( 1 m ) ( ζ + 2 ) ( ς 2 x ) ζ + 1 ( x ς 1 ) ζ + 1 ( ς 2 ς 1 ) + ( ς 2 x ) ζ + 1 ( x m ς 1 ) s ( ( 1 m ) ς 1 ) ζ + s + 2 B ( ζ + 2 , s ζ 2 ) + ( x ς 1 ) ζ + 1 ( ς 2 m x ) ζ + 2 B ( ζ + 2 , s + 1 ) ] .

2.12. Ostrowski-Type Fractional Integral Inequalities for n-Polynomial Exponentially s-Convex Functions

Now, we present some Ostrowski-type inequalities for differentiable exponentially s-convex functions.
Definition 19
([38]). Let s [ ln 2.4 , 1 ] . Then the real-valued function Π is an exponentially s-convex function if
Π ( t x + ( 1 t ) y ) ( e s t 1 ) Π ( x ) + ( e s ( 1 t ) 1 ) Π ( y ) ,
x , y I and t [ 0 , 1 ] .
Theorem 64
([38]). Let Π : I R R be a differentiable mapping on I , ς 1 , ς 2 I with ς 1 < ς 2 . If | Π | is an exponentially s-convex function on [ ς 1 , ς 2 ] for some s [ ln 2.4 , 1 ] , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( z ) d z | M ( ς 2 ς 1 ) [ ( x ς 1 ) 2 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 + 2 e s s 2 2 s 2 2 s 2 + ( ς 2 x ) 2 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 + 2 e s s 2 2 s 2 2 s 2 ] ,
for all x [ ς 1 , ς 2 ] .
Theorem 65
([38]). Let Π : I R R be a differentiable function on I , ς 1 , ς 2 I with ς 1 < ς 2 . If | Π | q is an exponentially s-convex function on [ ς 1 , ς 2 ] for some s [ ln 2.4 , 1 ] , q > 1 , 1 p + 1 q = 1 , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( z ) d z | 2 1 q M ( ς 2 ς 1 ) n q 1 p + 1 1 p ( x ς 1 ) 2 e s s 1 ) s 1 q + ( ς 2 x ) 2 e s s 1 ) s 1 q ,
for all x [ ς 1 , ς 2 ] .
Theorem 66
([38]). Let Π : I R R be a differentiable function on I , ς 1 , ς 2 I with ς 1 < ς 2 . If | Π | q , q 1 is an exponentially s-convex function on [ ς 1 , ς 2 ] for some s [ ln 2.4 , 1 ] , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( z ) d z | M ( ς 2 ς 1 ) 2 1 1 q [ ( x ς 1 ) 2 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 + 2 e s s 2 2 s 2 2 s 2 1 q + ( ς 2 x ) 2 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 + 2 e s s 2 2 s 2 2 s 2 1 q ] ,
for all x [ ς 1 , ς 2 ] .
Some enhancements of the Ostrowski-type inequality for differentiable n-polynomial exponentially s-convex functions are presented in the next theorems.
Definition 20
([39]). Let n N and s [ ln 2.4 , 1 ] . Then Π : I R R is an n-polynomial exponentially s-convex function if
Π ( t x + ( 1 t ) y ) 1 n i = 1 n ( e s t 1 ) i Π ( x ) + 1 n i = 1 n ( e s ( 1 t ) 1 ) i Π ( y ) ,
for all x , y I and t [ 0 , 1 ] .
Theorem 67
([39]). Let Π : I R R be a differentiable mapping on I , [ ς 1 , ς 2 ] I with ς 1 < ς 2 . If | Π | is an n-polynomial exponentially s-convex function on [ ς 1 , ς 2 ] for some s [ ln 2.4 , 1 ] , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( z ) d z | M ( ς 2 ς 1 ) n [ ( x ς 1 ) 2 i = 1 n 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 i + i = 1 n 2 e s s 2 2 s 2 2 s 2 i + ( ς 2 x ) 2 i = 1 n 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 i + i = 1 n 2 e s s 2 2 s 2 2 s 2 i ] ,
for all x [ ς 1 , ς 2 ] .
Theorem 68
([39]). Let Π be as in Theorem 67. If | Π | q is an n-polynomial exponentially s-convex function on [ ς 1 , ς 2 ] for some s [ ln 2.4 , 1 ] , q > 1 , and 1 p + 1 q = 1 , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then:
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( z ) d z | 2 1 q M ( ς 2 ς 1 ) n q 1 p + 1 1 p [ ( x ς 1 ) 2 i = 1 n e s s 1 ) s i 1 q + ( ς 2 x ) 2 i = 1 n e s s 1 ) s i 1 q ] ,
for all x [ ς 1 , ς 2 ] .
Theorem 69
([39]). Let Π be as in Theorem 67. If | Π | q , q 1 is an n-polynomial exponentially s-convex function on [ ς 1 , ς 2 ] for some s [ ln 2.4 , 1 ] , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then
| Π ( x ) 1 ς 2 ς 1 ς 1 ς 2 Π ( z ) d z | M ( ς 2 ς 1 ) n q 2 1 1 q [ ( x ς 1 ) 2 i = 1 n 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 i + i = 1 n 2 e s s 2 2 s 2 2 s 2 i 1 q + ( ς 2 x ) 2 i = 1 n 2 + 2 ( s 1 ) e s s 2 ) 2 s 2 i + i = 1 n 2 e s s 2 2 s 2 2 s 2 i 1 q ] ,
for all x [ ς 1 , ς 2 ] .
Theorem 70
([40]). Let the assumptions of this theorem be stated in Theorem 67. Then
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 [ J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M n ( ς 2 ς 1 ) [ ( x ς 1 ) ζ + 1 { i = 1 n B ( ζ + 1 , s ) Γ ( ζ + 1 ) ( s ) ζ s 1 ζ + 1 i i = 1 n ( B ( ζ + 1 , s ) Γ ( ζ + 1 ) ) e s s ζ + 1 + 1 ζ + 1 i } + ( ς 2 x ) ζ + 1 { i = 1 n B ( ζ + 1 , s ) Γ ( ζ + 1 ) ( s ) ζ s 1 ζ + 1 i i = 1 n ( B ( ζ + 1 , s ) Γ ( ζ + 1 ) ) e s s ζ + 1 + 1 ζ + 1 i } ] ,
for all x ( ς 1 , ς 2 ) .
Theorem 71
([40]). Let Π be as in Theorem 67. If | Π | q is an n-polynomial exponentially s-convex function on [ ς 1 , ς 2 ] for some s ( 0 , 1 ) , q > 1 , and 1 p + 1 q = 1 , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 [ J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | 2 1 q M n q ( ς 2 ς 1 ) 1 ζ p + 1 1 p [ ( x ς 1 ) ζ + 1 i = 1 n e s s 1 s i 1 q + ( ς 2 x ) ζ + 1 i = 1 n e s s 1 s i 1 q ] ,
for all x ( ς 1 , ς 2 ) .
Theorem 72
([40]). Let Π be as in Theorem 67. If | Π | q , q 1 is an n-polynomial exponentially s-convex function on [ ς 1 , ς 2 ] for some s ( 0 , 1 ) , Π L [ ς 1 , ς 2 ] and | Π ( x ) | M , for all x [ ς 1 , ς 2 ] , then:
| ( x ς 1 ) ζ + ( ς 2 x ) ζ ς 2 ς 1 Π ( x ) Γ ( ζ + 1 ) ς 2 ς 1 [ J x ζ Π ( ς 1 ) + J x + ζ Π ( ς 2 ) | M n q ( ς 2 ς 1 ) 1 ζ + 1 1 1 p [ ( x ς 1 ) ζ + 1 { i = 1 n B ( ζ + 1 , s ) Γ ( ζ + 1 ) ( s ) ζ s 1 ζ + 1 i i = 1 n ( B ( ζ + 1 , s ) Γ ( ζ + 1 ) ) e s s ζ + 1 1 ζ + 1 i } 1 q + ( ς 2 x ) ζ + 1 { i = 1 n B ( ζ + 1 , s ) Γ ( ζ + 1 ) ( s ) ζ s 1 ζ + 1 i i = 1 n ( B ( ζ + 1 , s ) Γ ( ζ + 1 ) ) e s s ζ + 1 1 ζ + 1 i } 1 q ] ,
for all x ( ς 1 , ς 2 ) .

3. Ostrowski-Type Inequalities for Katugampola Fractional Integral Operator

Here, we present some Ostrowski-type inequalities via the Katugampola fractional integral operator.
Definition 21
([41]). Let [ ς 1 , ς 2 ] R be a finite interval. Then, the left- and right-side Katugampola fractional integral of order ζ > 0 of Π X c p ( ς 1 , ς 2 ) are defined by
ρ I ς 1 + ζ Π ( x ) = ρ 1 ζ Γ ( ζ ) ς 1 x t ρ 1 ( x ρ t ρ ) 1 ζ Π ( t ) d t a n d ρ I ς 2 ζ Π ( x ) = ρ 1 ζ Γ ( ζ ) x ς 2 t ρ 1 ( t ρ x ρ ) 1 ζ Π ( t ) d t ,
with ς 1 < x < ς 2 and ρ > 0 , if the integrals exist. Here, X c p ( ς 1 , ς 2 ) , c R , 1 p denote the space of those complex-valued Lebesque measurable functions Π on [ ς 1 , ς 2 ] for which Π X c p < , where Π X c p = ς 1 ς 2 | t c Π ( t ) | p d t t 1 / p < for 1 p < and Π X c p = e s s sup x 1 t x 2 [ t c | Π ( t ) | ] , if p = .
Theorem 73
([42]). Let Π : [ ς 1 ρ , ς 2 ρ ] [ 0 , ) R be a differentiable function on ( ς 1 ρ , ς 2 ρ ) with ς 1 ρ < ς 2 ρ such that Π L [ ς 1 ρ , ς 2 ρ ] . If Π is h-convex on [ ς 1 ρ , ς 2 ρ ] and | Π ( x ρ ) | M , x [ ς 1 , ς 2 ] , then
| Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ρ I x ζ Π ( ς 1 ρ ) 2 ( x ρ ς 1 ρ ) ζ + ρ I x + ζ Π ( ς 2 ρ ) 2 ( ς 2 ρ x ρ ) ζ | M ρ ( ς 2 ρ ς 1 ρ ) 2 0 1 t ζ ρ + ρ 1 [ h ( t ρ ) + h ( 1 t ρ ) ] d t ,
with ζ , ρ > 0 and x ( ς 1 ρ , ς 2 ρ ) .
Theorem 74
([42]). Let Π be as in Theorem 73. If | Π | q , q > 1 is h-convex on [ ς 1 ρ , ς 2 ρ ] and | Π ( x ρ ) | M , x [ ς 1 , ς 2 ] , then:
| Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ρ I x ζ Π ( ς 1 ρ ) 2 ( x ρ ς 1 ρ ) ζ + ρ I x + ζ Π ( ς 2 ρ ) 2 ( ς 2 ρ x ρ ) ζ | M ρ ( ς 2 ρ ς 1 ρ ) 2 ( p ( ζ ρ + ρ 1 ) + 1 ) 1 p 0 1 [ h ( t ρ ) + h ( 1 t ρ ) ] d t 1 q ,
with ζ , ρ > 0 , x ( ς 1 ρ , ς 2 ρ ) and 1 p + 1 q = 1 .
Theorem 75
([42]). Let Π be as in Theorem 73. If | Π | q , q 1 is h-convex on [ ς 1 ρ , ς 2 ρ ] and | Π ( x ρ ) | M , x [ ς 1 , ς 2 ] , then:
| Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ρ I x ζ Π ( ς 1 ρ ) 2 ( x ρ ς 1 ρ ) ζ + ρ I x + ζ Π ( ς 2 ρ ) 2 ( ς 2 ρ x ρ ) ζ | M ρ ( ς 2 ρ ς 1 ρ ) 2 1 ρ ( ζ + 1 ) 1 1 q 0 1 t ζ ρ + ρ 1 [ h ( t ρ ) + h ( 1 t ρ ) ] d t 1 q ,
with ζ , ρ > 0 and x ( ς 1 ρ , ς 2 ρ ) .
Some Ostrowski-type inequalities pertaining to Katugampola fractional integral for s-Godunova-Levin functions are presented.
Theorem 76
([43]). Let Π : [ ς 1 ρ , ς 2 ρ ] [ 0 , ) R be a function which is differentiable on ( ς 1 ρ , ς 2 ρ ) with ς 1 < ς 2 such that Π L [ ς 1 , ς 2 ] . If | Π | is an s-Godunova-Levin function of the second kind on [ ς 1 ρ , ς 2 ρ ] and | Π ( x ρ ) | M , x [ ς 1 , ς 2 ] , then:
| ( x ρ ς 1 ρ ) ζ + ( ς 2 ρ x ) ζ ς 2 ς 1 Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ( ς 2 ς 1 ) ρ I x ζ Π ( ς 1 ρ + ρ I x ζ Π ( ς 1 ρ | M ( x ρ ς 1 ρ ) ζ + 1 + ( ς 2 ρ x ) ζ + 1 ς 2 ς 1 1 ζ + 1 s + Γ ( ζ + 1 ) Γ ( 1 s ) Γ ( ζ + 2 s ) ,
with ζ , ρ > 0 and x ( ς 1 ρ , ς 2 ρ ) .
Theorem 77
([43]). Let Π be as in Theorem 76. If | Π | q , q > 1 is an s-Godunova-Levin function of the second kind on [ ς 1 ρ , ς 2 ρ ] and | Π ( x ρ ) | M , x [ ς 1 , ς 2 ] , then:
| ( x ρ ς 1 ρ ) ζ + ( ς 2 ρ x ) ζ ς 2 ς 1 Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ( ς 2 ς 1 ) ρ I x ζ Π ( ς 1 ρ + ρ I x ζ Π ( ς 1 ρ | M ρ ( x ρ ς 1 ρ ) ζ + 1 + ( ς 2 ρ x ) ζ + 1 ( ς 2 ς 1 ) ( 1 + p ( ζ ρ + ρ 1 ) ) 1 p 1 1 ρ s 1 q ,
with ζ , ρ > 0 , x ( ς 1 ρ , ς 2 ρ ) and 1 p + 1 q = 1 .
Theorem 78
([43]). Let Π be as in Theorem 76. If | Π | q , q 1 is an s-Godunova-Levin function of the second kind on [ ς 1 ρ , ς 2 ρ ] and | Π ( x ρ ) | M , x [ ς 1 , ς 2 ] , then
| ( x ρ ς 1 ρ ) ζ + ( ς 2 ρ x ) ζ ς 2 ς 1 Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ( ς 2 ς 1 ) ρ I x ζ Π ( ς 1 ρ + ρ I x ζ Π ( ς 1 ρ | M ρ ( ζ ρ + ρ ) 1 1 q ( x ρ ς 1 ρ ) ζ + 1 + ( ς 2 ρ x ) ζ + 1 ς 2 ς 1 1 ρ ( ζ s + 1 ) + Γ ( ζ + 1 ) Γ ( 1 s ) ρ Γ ( ζ s + 2 ) 1 q ,
with ζ , ρ > 0 and x ( ς 1 ρ , ς 2 ρ ) .
Theorem 79
([43]). Let the assumptions of this theorem be as stated in Theorem 76. Then:
| Π ( x ρ ) ( ζ ρ + ρ 1 ) Γ ( ζ ) ρ 1 ζ ρ I x ζ Π ( ς 1 ρ 2 ( x ρ ς 1 ρ ) ζ + ρ I x + ζ Π ( ς 2 ρ 2 ( ς 2 ρ x ρ ) ζ |