On Opial-type inequality for a generalized fractional integral operator

Miguel Vivas-Cortez; Francisco Martínez; Juan E. Nápoles Valdes; Jorge E. Hernández

doi:10.1515/dema-2022-0149

Open Access Published by De Gruyter Open Access October 11, 2022

On Opial-type inequality for a generalized fractional integral operator

Miguel Vivas-Cortez , Francisco Martínez , Juan E. Nápoles Valdes and Jorge E. Hernández

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2022-0149

Abstract

This article is aimed at establishing some results concerning integral inequalities of the Opial type in the fractional calculus scenario. Specifically, a generalized definition of a fractional integral operator is introduced from a new Raina-type special function, and with certain results proposed in previous publications and the choice of the parameters involved, the established results in the work are obtained. In addition, some criteria are established to obtain the aforementioned inequalities based on other integral operators. Finally, a more generalized definition is suggested, with which interesting results can be obtained in the field of fractional integral inequalities.

Keywords: Opial inequality; fractional integral operator; fractional calculus

MSC 2010: 26D10; 26A33

1 Introduction

The study of integral inequalities has been a significant subfield of fractional calculus for decades, connecting with other areas such as differential equations, particularly those related to asymptotic properties of solutions of neutral differential equations, mathematical analysis, mathematical physics, convexity theory, numerical analysis applied to fractional partial differential equations, and discrete fractional calculus, as can be seen in [1,2,3, 4,5,6, 7,8,9]. One important type of integral inequalities consists of the so-called Opial inequality, which is related to the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [10]. This inequality has attracted continued interest from researchers and has proven to be very important in many situations, including the classical Riemann integral and fractional integrals, as shown in [11,12,13, 14,15]. The Opial inequality was given in [16] as follows:

Theorem 1.1

Let f ∈ C 1 ( [ 0 , h ] ) satisfy f ( 0 ) = f ( h ) = 0 and f ( x ) > 0 in ( 0 , h ) , then

(1) ∫ 0 h ∣ f ( x ) f ′ ( x ) ∣ d x ≤ h 4 ∫ 0 h ∣ f ′ ∣ 2 d x ,

where constant h ∕ 4 is the best choice.

It is known that, historically, from the correspondences between Leibniz and L’Hôpital, a field of research regarding what is currently known as fractional calculus is still in progress. One of the first most studied fractional integral operators is the Riemann-Liouville fractional integral operator of order μ > 0 , which is established in the left and right form as follows:

(2) ( ℐ a + μ φ ) ( x ) = 1 Γ ( μ ) ∫ a x ( x − ξ ) μ − 1 φ ( ξ ) d ξ , x > a ,

(3) ( ℐ b − μ φ ) ( x ) = 1 Γ ( μ ) ∫ x b ( ξ − x ) μ − 1 φ ( ξ ) d ξ , x < b ,

where φ is a function defined on x ∈ [ a , b ] .

Still, in 2014 and following this classic model, He defined the next model [17] as follows:

D t α f = 1 Γ ( n − α ) d n d t n ∫ t 0 t ( s − t ) n − α − 1 ( f 0 ( s ) − f ( s ) ) d s ,

where f 0 is a known function. Following the same steps to reach for the Riemann-Liouville fractional integral operator, we have the He fractional integral operator as follows:

(4) I t 0 α f = 1 Γ ( α ) ∫ t 0 t ( s − t ) α − 1 ( f 0 ( t ) − f ( t ) ) d s .

Both operators have shown their usefulness in applications to physics, for example, in the internal response to temperature given by porous concrete [18].

There are many directions to define fractional derivatives and fractional integrals, which are often related to or inspired by the Riemann-Liouville definitions, see, for example, [19,20,21], with reference to some general classes into which such fractional operators can be classified. In pure mathematics, we always consider the most general possible setting in which a specific behavior or result can be obtained. However, in applied mathematics, it is important to consider particular types of fractional calculus to the model of a given real-world problem.

Some of these definitions of fractional calculus have properties different from the standard Riemann-Liouville definition, and some of them can be used to improve the model of real-life data more effectively than the aforementioned model [22,23].

Special functions have many relations with fractional calculus, and the Mittag-Leffler function is a parti-cularly significant one in this area [24,25]. In addition, various novel definitions of integral and differential fractional operators have been introduced, and thorough investigations demonstrating their properties and applications have been conducted [26,27, 28,29,30, 31,32,33, 34,35].

In [36], Farid et al. established some results that relate the Opial inequality with fractional integral operators, that involve the Mittag-Leffler function, from some previously results shown in the following articles [37,38,39, 40,41].

In this article, we introduce a new, generalized definition that corresponds to a class of functions of the Raina-type, and with it certain parameterized fractional integral inequalities of the Opial type are established, which generalize others found in the aforementioned publications. The structure of this work is composed of four parts: an introductory section that provides the basic motivation for this development; a preliminaries section that supports the theoretical bases used; a main results section in which the results found are established; and finally a section of conclusions.

2 Preliminaries

It is well known that Mittag-Leffler introduced the following special function in 1903 (see [24]):

(5) E α ( z ) = ∑ n = 0 ∞ z n Γ ( α n + 1 )

for α ≥ 0 , z ∈ C , and Γ the Gamma function.

A generalization was given by Wiman in [25] as follows:

E α , β ( z ) = ∑ n = 0 ∞ z n Γ ( α n + β )

for α , β ∈ C with ℛ e ( α ) > 0 , ℛ e ( β ) > 0 , and z ∈ C .

In addition, Prabhakar in [26] gave a more generalized definition as follows:

(6) E α , β γ ( z ) = ∑ n = 0 ∞ ( γ ) n Γ ( α n + β ) n ! z n

for α , β , γ ∈ C with ℛ e ( α ) > 0 , ℛ e ( β ) > 0 , ( ℛ ) ( γ ) > 0 , and z ∈ C , where ( γ ) n is the Pochhammer symbol. In addition, the left and right fractional integral operators were given as follows:

(7) I α , β , w , a + γ φ ( x ) = ∫ a x ( x − ξ ) β − 1 E α , β γ [ w ( x − ξ ) α ] φ ( ξ ) d ξ , I α , β , w , b − γ φ ( x ) = ∫ x b ( ξ − x ) β − 1 E α , β γ [ w ( ξ − x ) α ] φ ( ξ ) d ξ .

Later, Salim and Faraj in [27] proposed the following model:

(8) E α , β , p γ , δ , q ( x ) = ∑ n = 0 ∞ ( γ ) q n Γ ( α n + β ) ( δ ) p n x k ,

where α , β , γ , δ ∈ C ; min { R e ( α ) , ℛ e ( β ) , ℛ e ( γ ) , ℛ e ( δ ) } > 0 ; p , q > 0 , and q ≤ ℛ e ( α ) + p .

In the same article, the left and right fractional integral operators were defined as follows:

(9) ε α , β , w , p , a + γ , δ , q φ ( x ) = ∫ a x ( x − ξ ) β − 1 E α , β , q γ , δ , p [ w ( x − ξ ) α ] φ ( ξ ) d ξ , ε α , β , w , p , b − γ , δ , q φ ( x ) = ∫ x b ( ξ − x ) β − 1 E α , β , q γ , δ , p [ w ( ξ − x ) α ] φ ( ξ ) d ξ .

In [28], Raina introduced a class of functions defined formally by

(10) ℱ ρ , λ σ ( x ) = ℱ ρ , λ σ ( 0 ) , σ ( 1 ) , … ( x ) = ∑ n = 0 ∞ σ ( n ) Γ ( ρ n + λ ) x n ,

where ρ , λ > 0 , ∣ x ∣ < R , ( R is the set of real numbers), and σ = ( σ ( 1 ) , … , σ ( k ) , … ) is a bounded sequence of positive real numbers. Observe that:

If we select ρ = 1 , λ = 0 , and σ ( n ) = ( ( μ ) k ( β ) k ∕ ( γ ) k ) for k = 0 , 1 , 2 , … . in (10), then the classical Hypergeometric Function is obtained, that is,
ℱ ρ , λ σ ( x ) = ∑ k = 0 ∞ ( μ ) k ( β ) k ( γ ) k k ! x k ,
where μ , β , and γ are parameters that can take arbitrary complex or real values such that γ ≠ 0 , − 1 , − 2 , … , and ( a ) k is the Pochhammer symbol given by
( a ) n = Γ ( a + n ) Γ ( a ) = a ( a + 1 ) ⋯ ( a + n − 1 ) , k = 0 , 1 , … ,
and restrict its domain to ∣ x ∣ ≤ 1 with x ∈ C .
If we take σ ( n ) = ( 1 , 1 , 1 , … ) with ρ = α , ( ℛ e ( α ) > 0 ) , λ = 1 , and restricting its domain to x ∈ C in (10), then we obtain the classical Mittag-Leffler function (5).
If we take σ ( n ) = ( η ) n n ! , ρ = α , and λ = β , we recover the Prabhakar extension model of the Mittag-Leffler function (6).
If we take σ ( k ) = ( γ ) q k ( δ ) p k , ρ = α , and λ = β in (10), we obtain the extension of the Mittag-Leffler function (8).

Making use of (10) and differentiating the right side termwise (which is permissible provided the series converges uniformly in any compact set of C ), we readily obtain

d d x n x λ − 1 ℱ ρ , λ σ ( w x ρ ) = x λ − n − 1 ℱ ρ , λ − n σ ( w x ρ ) ,

where ρ , λ , w ∈ C ( ℛ e ( ρ ) > 0 , ℛ e ( λ ) > 0 ) and n ∈ N .

Similarly, we can obtain the following result:

∫ 0 x … ∫ 0 x ︸ n times t λ − 1 ℱ ρ , λ σ ( w t ρ ) ( d t ) n = x λ + n − 1 ℱ ρ , λ + n σ ( w x ρ ) ,

where ρ , λ , w ∈ C ( ℛ e ( ρ ) > 0 , ℛ e ( λ ) > 0 ) and n ∈ N .

Meantime, Raina’s fractional model is proposed in [28,29] and is defined similarly to the Riemann-Liouville fractional integral (2) but with a modified special function in the kernel. This is defined as follows.

Definition 2.1

For any function φ which is L 1 ( [ a , b ] ) , the left and right Raina’s fractional integral operators applied to φ are defined by the following integral transforms, for λ , ρ > 0 , w ∈ R :

(11) ( J ρ , λ , a + ; w σ φ ) ( x ) = ∫ a x ( x − ξ ) λ − 1 ℱ ρ , λ σ [ w ( x − ξ ) ρ ] φ ( ξ ) d ξ , ( x > a )

and

(12) ( J ρ , λ , b − ; w σ φ ) ( x ) = ∫ x b ( ξ − x ) λ − 1 ℱ ρ , λ σ [ w ( ξ − x ) ρ ] φ ( ξ ) d ξ , ( x < b ) ,

where φ is such that the integral on the right side exits.

It is easy to verify that J ρ , λ , a + ; w σ φ and J ρ , λ , b − ; w σ φ are bounded integral operators on L p ( a , b ) , ( 1 ≤ p ≤ ∞ ) , if

ℳ = ℱ ρ , λ + 1 σ [ w ( b − a ) ρ ] < ∞ .

In fact, for φ ∈ L p ( a , b ) , we have

∥ J ρ , λ , b 1 + ; w σ φ ∥ p ≤ ℳ ∥ φ ∥ p and ∥ J ρ , λ , b 2 − ; w σ φ ∥ p ≤ ℳ ∥ φ ∥ p ,

where

∥ φ ∥ p = ∫ a b ∣ φ ( x ) ∣ p d x 1 ∕ p .

The importance of the Raina’s operators stems indeed from their generality. That is, by specifying the coefficient σ ( n ) , we can obtain many useful fractional integral operators as follows:

If we choose λ = α , σ ( 0 ) = 1 , σ ( n ) = 0 for n ≠ 0 , and w = 0 in Definition 2.1, we can deduce the left and right Riemann-Liouville fractional integrals (2) and (3). Also, with the same choices, we obtain the He fractional integral (4).
If we choose σ ( n ) = ( η ) n n ! , ρ = α , and λ = β in Definition 2.1, we obtain the Prabhakar left and right fractional integrals (7).
If we choose σ ( n ) = ( γ ) q k ( δ ) p k , ρ = α , and λ = β in Definition 2.1, we obtain the Salim-Faraj fractional integral (9).

There exist many integral inequalities related to the Raina fractional operators; the reader may refer to [30,31, 32,33], where the aforementioned fractional integral operator was improved.

The following results are necessary for the development of this work. In [37], the class U ( v , K ) , which contains all the functions u : [ a , b ] → R that admit the representation

u ( x ) = ∫ a x K ( x , t ) v ( t ) d t ,

where v is a continuous function such that v ( x ) > 0 implies u ( x ) > 0 with K as an arbitrary non-negative kernel. With this definition, the authors established the following theorem.

Theorem 2.1

Let u 1 ∈ U ( v 1 , K ) and u 2 ∈ U ( v 2 , K ) with v 2 ( x ) > 0 for every x ∈ [ a , b ] . Furthermore, let φ ( u ) be convex, non-negative and increasing for u ≥ 0 , f ( u ) be convex for u ≥ 0 , and f ( 0 ) = 0 . If f is a differentiable function and M = max K ( x , t ) , then

M ∫ a b v 2 ( t ) φ v 1 ( t ) v 2 ( t ) f ′ u 2 ( t ) φ u 1 ( t ) u 2 ( t ) d t ≤ f M ∫ a b v 2 ( t ) φ v 1 ( t ) v 2 ( t ) d t .

In [39], Andrić et al. gave an extension of the previous theorem.

Theorem 2.2

M ∫ a b v 2 ( t ) φ v 1 ( t ) v 2 ( t ) f ′ u 2 ( t ) φ u 1 ( t ) u 2 ( t ) d t ≤ f M ∫ a b v 2 ( t ) φ v 1 ( t ) v 2 ( t ) d t ≤ 1 b − a ∫ a b f M ( b − a ) v 2 ( t ) φ v 1 ( t ) v 2 ( t ) d t .

In [40], Pećarić et al. established the following result.

Theorem 2.3

Let φ : [ 0 , ∞ ) → R be a differentiable function such that for q > 1 , the function φ ( x 1 ∕ q ) is convex and φ ( 0 ) = 0 . Let u ∈ U ( v , K ) , where

∫ a x ( K ( x , t ) ) p d t 1 ∕ p ≤ M

and 1 ∕ p + 1 ∕ q = 1 . Then

∫ ∣ u ( x ) ∣ 1 − 1 ∕ q φ ′ ( ∣ u ( x ) ∣ ) ∣ v ( x ) ∣ q d x ≤ q M q φ M ∫ a b ∣ v ( x ) ∣ q d x 1 ∕ q .

The reverse inequality holds if φ ( x 1 ∕ q ) is concave.

Also, an extension of the above theorem is presented in [41].

Theorem 2.4

Let φ : [ 0 , ∞ ) → R be a differentiable function such that for q > 1 , the function φ ( x 1 ∕ q ) is convex and φ ( 0 ) = 0 . Let u ∈ U ( v , K ) , where

∫ a x ( K ( x , t ) ) p d t 1 ∕ p ≤ M

and 1 ∕ p + 1 ∕ q = 1 . Then

∫ ∣ u ( x ) ∣ 1 − q φ ′ ( ∣ u ( x ) ∣ ) ∣ v ( x ) ∣ q d x ≤ q M q φ M ∫ a b ∣ v ( x ) ∣ q d x 1 ∕ q ≤ q M q ( b − a ) ∫ a b φ ( ( b − a ) 1 ∕ q M ∣ v ( x ) ∣ ) d x .

The reverse inequality holds if φ ( x 1 ∕ q ) is concave.

In [38], Andrić studied the following functional:

Ψ φ ( u , v ) = q M q ( b − a ) ∫ a b φ ( ( b − a ) 1 ∕ q M ∣ v ( x ) ∣ ) d x − ∫ ∣ u ( x ) ∣ 1 − q φ ′ ( ∣ u ( x ) ∣ ) ∣ v ( x ) ∣ q d x ,

establishing some general properties and the following theorems.

Theorem 2.5

Let φ : [ 0 , ∞ ) → R be a differentiable function such that for q > 1 , the functions φ ( x 1 ∕ q ) are convex and φ ( 0 ) = 0 . Let u ∈ U ( v , K ) , where

∫ a x ( K ( x , t ) ) p d t 1 ∕ p ≤ M

and 1 ∕ p + 1 ∕ q = 1 . If φ ∈ C 2 ( I ) , where I ⊂ ( 0 , ∞ ) is a compact interval, then there exists ξ ∈ I such that

Ψ φ 1 ( u , v ) = ξ φ 1 ″ ( ξ ) − ( q − 1 ) φ 1 ′ ( ξ ) 2 q ξ 2 q − 1 ( b − a ) M q ∫ a b ∣ v ( x ) ∣ 2 q d x − 2 ∫ a b ∣ u ( x ) ∣ q ∣ v ( x ) ∣ q d x .

Theorem 2.6

Let φ 1 , φ 2 : [ 0 , ∞ ) → R be differentiable functions such that for q > 1 , the function φ i ( x 1 ∕ q ) is convex and φ i ( 0 ) = 0 for i = 1 , 2 . Let u ∈ U ( v , K ) , where

∫ a x ( K ( x , t ) ) p d t 1 ∕ p ≤ M

and 1 ∕ p + 1 ∕ q = 1 . If φ 1 , φ 2 ∈ C 2 ( I ) , where I ⊂ ( 0 , ∞ ) is a closed interval and

( b − a ) M q ∫ a b ∣ v ( x ) ∣ 2 q d x − 2 ∫ a b ∣ u ( x ) ∣ q ∣ v ( x ) ∣ q d x ≠ 0 ,

then there exists ξ ∈ I such that

Ψ φ 1 ( u , v ) Ψ φ 2 ( u , v ) = ξ φ 1 ″ ( ξ ) − ( q − 1 ) φ 1 ′ ( ξ ) ξ φ 2 ″ ( ξ ) − ( q − 1 ) φ 2 ′ ( ξ ) .

Theorem 2.7

Let u 1 ∈ U ( v 1 , K ) , u 2 ∈ U ( v 2 , K ) , and v 2 ( x ) > 0 for all x ∈ [ a , b ] . Furthermore, let ϕ ( x ) be convex, non-negative, and increasing for x ≥ 0 , f ( x ) be convex for x ≥ 0 , and f ( 0 ) = 0 . Let α , β , k , l , γ , ω > 0 such that k < l + α and β > 1 , then we have

E α , β , l γ , δ , k ( w ( b − a ) α ) ( b − a ) β − 1 ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) f ′ ( ε α , β , w , l , a + γ , δ , k v 2 ) ( x ) ϕ ( ε α , β , w , l , a + γ , δ , k v 1 ) ( x ) ( ε α , β , w , l , a + γ , δ , k v 2 ) ( x ) d x ≤ f E α , β , l γ , δ , k ( w ( b − a ) α ) ( b − a ) β − 1 ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x ,

where E α , β , l γ , δ , k and ε α , β , w , l , a + γ , δ , k are the generalized Mittag-Leffler function and the integral fractional operator introduced by Salim and Faraj in [27], respectively.

With this preliminary knowledge, we present the results obtained in Section 3.

3 Main results

Before presenting some results that involve Opial-type inequality, we will give the following definition regarding the conformable fractional integral of the Raina-type.

Definition 3.1

Let a , b > 0 and b > a . For any function φ ∈ L 1 ( [ a , b ] ) , the following fractional integral operators for λ , ρ > 0 , w ∈ R , ϖ ∈ ( 0 , 1 ] , ϑ ∈ R , with ϑ + ϖ ≠ 0 and σ a bounded arbitrary sequence of real (or complex) numbers, are defined:

(13) ( J ρ , λ , a + ; w σ ϖ ϑ φ ) ( x ) = ∫ a x x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] φ ( ξ ) d ξ ,

in the case of x > a , and

(14) ( J ρ , λ , b − ; w σ ϖ ϑ φ ) ( x ) = ∫ x b ξ ϑ + ϖ − x ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( ξ ϑ + ϖ − x ϑ + ϖ ) ρ ] φ ( ξ ) d ξ ,

in the case of x < b , where φ is such that the integral on the right side exits.

Remark 3.1

Some fractional integral operators can be obtained from a specific choice of the parameters introduced in Definition 3.1:

Letting ρ = 1 , w = 1 , and σ ( 1 ) = 0 and σ ( n ) = 0 for n ≠ 0 , we have the left and right generalized fractional conformable integral operators introduced by Khan and Khan in [34] and cited by Nisar et al. in [35] as follows:
( ℐ a + λ ϖ ϑ φ ) ( x ) = 1 Γ ( λ ) ∫ a x x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 φ ( ξ ) d ξ , ( x > a , a > 0 ) ,
and
( ℐ b − λ ϖ ϑ φ ) ( x ) = 1 Γ ( λ ) ∫ x b ξ ϑ + ϖ − x ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 φ ( ξ ) d ξ , ( x < b , b > 0 ) ;
also, if ϑ = 0 , then we obtain the Katugampola conformable fractional integral operators [21] as follows:
( ℐ a + λ ϖ φ ) ( x ) = 1 Γ ( λ ) ∫ a x x ϖ − ξ ϖ ϖ λ − 1 ξ ϖ − 1 φ ( ξ ) d ξ , ( x > a , a > 0 ) ,
and
( ℐ b − λ ϖ φ ) ( x ) = 1 Γ ( λ ) ∫ x b ξ ϖ − x ϖ ϖ λ − 1 ξ ϖ − 1 φ ( ξ ) d ξ , ( x < b , b > 0 ) ;
and with the choice ϖ = 1 , is easy to obtain, from (13) and (14), the left and right Riemann-Liouville fractional integral.
Similarly, with the choice of ϖ = 1 , ϑ = 0 , σ ( k ) = ( η ) k k ! , ρ = μ , and λ = β , we obtain the Prabhakar fractional integral (7), and if we take σ ( k ) = ( γ ) q k ( δ ) p k , ρ = μ , and λ = β , then we obtain the Salim fractional integral (9).

Note that if ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] < ∞ and φ ∈ L p ( 0 , 1 ) with 1 ≤ p < ∞ , then

x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] ≤ b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] = ℳ ,

∥ ( J ρ , λ , a + ; w σ ϖ ϑ φ ) ∥ p ≤ ℳ ∥ φ ∥ p

and, similarly,

∥ ( J ρ , λ , b − ; w σ ϖ ϑ φ ) ∥ p ≤ ℳ ∥ φ ∥ p .

Throughout this study, we suppose that σ is an arbitrary bounded sequence real number.

Theorem 3.1

Let v 1 , v 2 : [ a , b ] → R be a continuous function with v 2 ( x ) ≥ 0 , φ convex, non-negative, and increasing function for x ≥ 0 , and f convex for x ≥ 0 with f ( 0 ) = 0 . Let λ , ρ > 0 , w ∈ R , and ϖ ∈ ( 0 , 1 ] , ϑ ∈ R with ϑ + ϖ ≠ 0 and σ an arbitrary bounded sequence of real (or complex) numbers, then we have

(15) M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) f ′ ( J ρ , λ , a + ; w σ ϖ ϑ v 2 ) ( x ) ϕ ( J ρ , λ , a + ; w σ ϖ ϑ v 1 ) ( x ) ( J ρ , λ , a + ; w σ ϖ ϑ v 2 ) ( x ) d x ≤ f M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] .

Proof

Let us define

K ( ξ , x ) = x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] a ≤ ξ ≤ x 0 x < ξ ≤ b ,

u 1 ( x ) = ( J ρ , λ , a + ; w σ ϖ ϑ v 1 ) ( x ) = ∫ a x x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] v 1 ( ξ ) d ξ ,

and

u 2 ( x ) = ( J ρ , λ , a + ; w σ ϖ ϑ v 2 ) ( x ) = ∫ a x x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] v 2 ( ξ ) d ξ .

Furthermore, it is not difficult to observe that

∑ k = 0 ∞ σ ( k ) Γ ( ρ k + λ ) w k ( x ϑ + ϖ − ξ ϑ + ϖ ) k ρ ≤ ∑ k = 0 ∞ σ ( k ) Γ ( ρ k + λ ) w k ( b ϑ + ϖ − a ϑ + ϖ ) k ρ = ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ,

x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ≤ b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ,

and

(16) K ( ξ , x ) ≤ b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] .

Letting

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ,

we can apply Theorem 2.1 and obtain inequality (15).□

Remark 3.2

With the same assumptions of the previous theorem and letting in the proof

K ( ξ , x ) = 0 a ≤ ξ ≤ x ξ ϑ + ϖ − x ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( ξ ϑ + ϖ − x ϑ + ϖ ) ρ ] x < ξ ≤ b ,

we have

M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) f ′ ( J ρ , λ , b − ; w σ ϖ ϑ v 2 ) ( x ) ϕ ( J ρ , λ , b − + ; w σ ϖ ϑ v 1 ) ( x ) ( J ρ , λ , b − ; w σ ϖ ϑ v 2 ) ( x ) d x ≤ f M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x .

This is the case for the right-side fractional integral operator (14).

Remark 3.3

With a suitable choice of the parameters, we obtain the following:

M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) f ′ ( ( ℐ a + λ ϖ ϑ v 2 ) ( x ) ) ( x ) ϕ ( ( ℐ a + λ ϖ ϑ v 1 ) ( x ) ) ( x ) ( ( ℐ a + λ ϖ ϑ v 2 ) ( x ) ) ( x ) d x ≤ f M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x

and

M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) f ′ ( ( ℐ b − λ ϖ ϑ v 2 ) ( x ) ) ( x ) ϕ ( ( ℐ b − λ ϖ ϑ v 1 ) ( x ) ) ( x ) ( ( ℐ b − λ ϖ ϑ v 2 ) ( x ) ) ( x ) d x ≤ f M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1

for ρ = 1 , w = 1 , and σ ( 1 ) = 0 , σ ( n ) = 0 for n ≠ 0 ; this is the case for the left generalized fractional conformable integral operators introduced in [34]. Similarly, with ϑ = 0 , then we obtain the inequality (15) for the Katugampola conformable fractional integral operators. Furthermore, with ϖ = 1 , we recover the inequality for the Riemann-Liouville fractional integral operator. Following the choices shown in #2, Remark 3.1, Theorem 2.7 for the Prabhakar and Salim fractional integral operators [37] is obtained.

Theorem 3.2

With the same assumptions of Theorem 3.1, we have

(17) M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) f ′ ( J ρ , λ , a + ; w σ ϖ ϑ v 2 ) ( x ) ϕ ( J ρ , λ , a + ; w σ ϖ ϑ v 1 ) ( x ) ( J ρ , λ , a + ; w σ ϖ ϑ v 2 ) ( x ) d x ≤ f M ∫ a b v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x ≤ 1 b − a ∫ a b f M v 2 ( x ) ϕ v 1 ( x ) v 2 ( x ) d x ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] .

Proof

Using the same methodology in Theorem 3.2 and applying Theorem 2.2, we obtain (17).□

Remark 3.4

Following the comments in Remark 3.3, we obtain inequality (17) for the fractional integral operator defined by Khan, Katugampola, Riemann-Liouville, Prabhakar, and Salim.

Theorem 3.3

Let φ : [ 0 , ∞ ) → R be a differentiable function such that the function φ ( x 1 ∕ q ) is convex and φ ( 0 ) = 0 , where q > 1 and 1 ∕ p + 1 ∕ q = 1 . Furthermore, let λ , ρ > 0 , w ∈ R , and ϖ ∈ ( 0 , 1 ] , ϑ ∈ R , ϑ + ϖ ≠ 0 , and σ an arbitrary bounded sequence of real (or complex) numbers, then we have

∫ ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ 1 − q φ ′ ( ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ ) ∣ v ( x ) ∣ q d x ≤ q M q φ M ∫ a b ∣ v ( x ) ∣ q d x 1 ∕ q ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q .

Proof

Let us define

K ( ξ , x ) = x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] , a ≤ ξ ≤ x 0 , x < ξ ≤ b

and

u ( x ) = ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) = ∫ a x x ϑ + ϖ − ξ ϑ + ϖ ϑ + ϖ λ − 1 ξ ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( x ϑ + ϖ − ξ ϑ + ϖ ) ρ ] v ( ξ ) d ξ .

From (16), we have

∫ a b ( K ( x , t ) ) p d t 1 ∕ p ≤ b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ∫ a b d x 1 ∕ p ≤ b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q .

Setting

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q ,

we apply Theorem 2.3, and therefore the desired results follows.□

Theorem 3.4

∫ a b ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ 1 − q φ ′ ( ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ ) ∣ v ( x ) ∣ q d x ≤ q M q φ M ∫ a b ∣ v ( x ) ∣ q d x 1 ∕ q ≤ q M q ( b − a ) ∫ a b φ ( ( b − a ) 1 ∕ q M ∣ v ( x ) ∣ ) d x ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q .

Proof

Following the previous proof and using Theorem 2.4, we obtain the desired result.□

The following result is obtained by using the function φ ( x ) = x p + q .

Corollary 3.1

Let q > 1 with 1 ∕ p + 1 ∕ q = 1 , λ , ρ > 0 , w ∈ R , and ϖ ∈ ( 0 , 1 ] , ϑ ∈ R , ϑ + ϖ ≠ 0 , and σ a bounded arbitrary sequence of real (or complex) numbers, then we have

∫ ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ p ∣ v ( x ) ∣ q d x ≤ q M q φ M ∫ a b ∣ v ( x ) ∣ q d x 1 ∕ q ≤ q M q ( b − a ) ∫ a b φ ( ( b − a ) 1 ∕ q M ∣ v ( x ) ∣ ) d x ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q .

Let us denote

Ψ φ ( ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) , v ) = q M q ( b − a ) ∫ a b φ ( ( b − a ) 1 ∕ q M ∣ v ( x ) ∣ ) d x − ∫ ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ 1 − q φ ′ ( ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ) ∣ v ( x ) ∣ q d x ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q .

Theorem 3.5

Let φ : [ 0 , ∞ ) → R be a differentiable function such that the function φ ( x 1 ∕ q ) is convex and φ ( 0 ) = 0 , where q > 1 with 1 ∕ p + 1 ∕ q = 1 . Let λ , ρ > 0 , w ∈ R , and ϖ ∈ ( 0 , 1 ] , ϑ ∈ R > 0 , ϑ + ϖ ≠ 0 , and σ a bounded arbitrary sequence of real (or complex) numbers, then we have the following equality:

Ψ φ ( ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) , v ) = ξ φ ″ ( ξ ) − ( q − 1 ) φ ′ ( ξ ) 2 q ξ 2 q − 1 ( b − a ) M q ∫ a b ∣ v ( x ) ∣ 2 q d x − 2 ∫ a b ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ q ∣ v ( x ) ∣ q d x .

Proof

From Theorem 3.3, we have

∫ a b ( K ( x , t ) ) p d t 1 ∕ p ≤ b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q .

Using

u ( x ) = ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x )

and

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q

in Theorem 2.5, we obtain the desired result.□

Theorem 3.6

Let λ , ρ > 0 , w ∈ R , and ϖ ∈ ( 0 , 1 ] , ϑ ∈ R > 0 , ϑ + ϖ ≠ 0 , and σ a bounded arbitrary sequence of real (or complex) numbers. Let φ 1 , φ 2 : [ 0 , ∞ ) → R be differentiable functions such that for q > 1 , the function φ i ( x 1 ∕ q ) is convex and φ i ( 0 ) = 0 for i = 1 , 2 , where q > 1 with 1 ∕ p + 1 ∕ q = 1 . Furthermore, if φ 1 , φ 2 ∈ C 2 ( I ) , where I ⊂ ( 0 , ∞ ) is a closed interval and

( b − a ) M q ∫ a b ∣ v ( x ) ∣ 2 q d x − 2 ∫ a b ∣ ( J ρ , λ , a + ; w σ ϖ ϑ v ) ( x ) ∣ q ∣ v ( x ) ∣ q d x ≠ 0 ,

where

M = b ϑ + ϖ − a ϑ + ϖ ϑ + ϖ λ − 1 b ϑ + ϖ − 1 ℱ ρ , λ σ [ w ( b ϑ + ϖ − a ϑ + ϖ ) ρ ] ( b − a ) 1 − 1 ∕ q ,

then there exists ξ ∈ I such that

Ψ φ 1 ( u , v ) Ψ φ 2 ( u , v ) = ξ φ 1 ″ ( ξ ) − ( q − 1 ) φ 1 ′ ( ξ ) ξ φ 2 ″ ( ξ ) − ( q − 1 ) φ 2 ′ ( ξ ) .

Proof

Similar to the proof of Theorem 3.5 and using Theorem 2.6, we obtain the result.□

Remark 3.5

Similar results can be obtained for the fractional operators described in Remark 3.1 comment by using the parameter values shown there.

4 Conclusion

In this work, we have introduced a new generalized definition of a fractional integral operator of Raina’s type, and from this, some theorems related to the Opial inequality were established. With a convenient choice of the parameters involved in the definition introduced, results previously shown in articles cited in the Section 1 are found.

We think that using the same methodology used in this work, it is possible to establish similar results for the following fractional integral operator model:

( J ρ , λ , a + ; w σ , k , g φ ) ( x ) = ∫ a x g ′ ( t ) ( g ( x ) − g ( t ) ) 1 − λ ∕ t ℱ ρ , λ σ , k [ w ( g ( x ) − g ( t ) ) ρ ] φ ( t ) d t ,

where k > 0 and g is a real valued, positive, and increasing monotone function defined on [ a , b ] , with continuous derivative g ′ on ( a , b ) , where

ℱ ρ , λ σ , k ( x ) = ℱ ρ , λ σ ( 0 ) , σ ( 1 ) , … ( x ) = ∑ n = 0 ∞ σ ( n ) k Γ k ( ρ k n + λ ) x n .

Acknowledgments

The authors thank the Dirección de Investigación from Pontificia Universidad Católica del Ecuador, the Universidad Politécnica de Cartagena (España), and the Concejo de Desarrollo Científico, Humanístico y Tecnológico from Universidad Centroccidental Lisandro Alvarado (Venezuela) for their contributions to the preparation of this work.

Funding information: This work received financial support from Dirección de Investigación from Pontificia Universidad Católica del Ecuador under the project entitled “Resultados cualitativos de ecuaciones diferenciales fraccionarias locales y desigualdades integrales,” code 070-UIO-2022.
Author contributions: All authors contributed equally to and approved the final manuscript.
Conflict of interest: The authors declare that there is no conflict of interest regarding the publication of this article.

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Received: 2022-03-17

Revised: 2022-06-02

Accepted: 2022-07-07

Published Online: 2022-10-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

On Opial-type inequality for a generalized fractional integral operator

Abstract

1 Introduction

Theorem 1.1

2 Preliminaries

Definition 2.1

Theorem 2.1

Theorem 2.2

Theorem 2.3

Theorem 2.4

Theorem 2.5

Theorem 2.6

Theorem 2.7

3 Main results

Definition 3.1

Remark 3.1

Theorem 3.1

Proof

Remark 3.2

Remark 3.3

Theorem 3.2

Proof

Remark 3.4

Theorem 3.3

Proof

Theorem 3.4

Proof

Corollary 3.1

Theorem 3.5

Proof

Theorem 3.6

Proof

Remark 3.5

4 Conclusion

Acknowledgments

References

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